Would a fair distribution of wealth from the super rich increase the purchasing power/life quality of the average person? I will start by saying that I do not have any background in finance and that the following question might be very much naive, but it has been bugging me for a while now. The general narrative seems to be  that redistribution of wealth from the super rich (top 1%) would benefit the other 99% and increase their quality of life. In the answer to this question, it is calculated that in case of redistribution from the top 1%, everybody among the bottom 99% would receive around $111,000. However, this answer does not really go into detail on what would happen in terms of purchasing power if everyone in the bottom 99% would get said amount. Specifically, my question is the following (and is, to first degree, purely of abstract nature). Let us consider a closed economic system in which most of the population is similarly wealthy, but a very tiny fraction owns a large amount of money. For simplicity, let us assume that we have a city with 1,000,0000 inhabitants from which 999,999 inhabitants are similarly wealthy, while there is 1 inhabitant who owns 90% of the wealth. Let us assume the wealth this one person owns is purely in the form of cash and he basically just hoards his money, so that this money is virtually non-existent in our system. Now assume that his money gets equally distributed between the inhabitants. Would this really make any difference to the purchasing power of the inhabitants and thus to their life quality? Wouldn't prices just simply rise so that purchasing power would stay roughly constant? The question boils down to two main assumptions:

The number of super rich people is much smaller than the number of average people
The money of the super rich is hoarded and virtually non-existent in our closed economic system

If my reasoning is correct with those assumptions, how are these assumption distorted in the real world?
tl;dr: In the hypothetical you set in the body of your question redistribution cannot help poor. However, this is not because redistribution could not significantly raise the welfare of the poor but rather because in your question 'the rich' actually dont have any resources to share with the rest. In fact the way how you set up your hypothetical example what you call 'redistribution' is equivalent to monetary expansion.
However, in real life redistribution, even redistribution of money, is not equivalent to monetary expansion and can help to raise material welfare of people especially of the poor and to the extent quality of life depends on material welfare their quality of life as well.

Full Answer:
The full answer will be structured in three parts. I will first try to answer your question in body of the text, then explain the problems with your assumptions and finally try to answer question in your title with small changes that will be still true to the spirit of the question and make it actually answerable in objective fashion (I do this because I think that most of the attention this question is driving comes from the question in its title).
PART I: Answer to the Question in the Body:
In order to answer your Q lets try to model it. In your example you talk about hypothetical economy when nobody works there are just people with some given wealth so we are in some endowment economy where people are endowed with some output $Y$ which constitutes their 'wealth', but rich people have only cash/money $M_r$. What even more you assume that the cash was not used in anyway so for all practical purposes it simply is not even part of the economy. Moreover, as per the question I will assume the money are equally distributed among everyone.
Well taking assumptions above seriously we can just analyze the effect of such money transfer the same way as just an expansion of money supply using simple equation of exchange (this is not normally appropriate model for evaluating redistribution but in your case redistribution is just monetary expansion in disguise and this model is a simple textbook example of a model that can be used to analyse effect of monetary expansion):
$$MV=PY$$
where, $V$ and $P$ which were not yet introduced  are velocity of money and price level respectively. In this case $Y$ is fixed as there is no production just everyone has some certain level of wealth. Hence when $M$ increases by adding the $M_r$ either $P$ increases or $V$ must drop (which would happen if the money people get from the rich end up 'buried' and not used), or some combination of thereof.
In either case whether prices or velocity changes people cant be made better off by the transfer of cash in your hypothetical scenario as in your scenario the rich $1\%$ really has nothing of value to give to the $99\%$. Their endowments $Y$ remain fixed. Again, I would normally never recommend using this model for analyzing redistribution but in your case you are not really talking about redistribution.
PART II: Problems With Your Question:
The main problem to your question is that if really taken to its logical conclusion it boils down to the following. Does creating more money make people any richer? Well answer to that is no. What makes people rich is the amounts of goods and services they have access to. In fact from an economic perspective the $1\%$ of population in your question must be miserable bunch because by your assumption they only have money while the other $99\%$ actually have some other 'wealth' which presumably in your question implies some goods and services to enjoy.
However, the above situation is clearly absurd and does not correspond to reality. Rich people do not sit upon piles of sterile cash that is not used in economy. If their money circulates in the economy either because the rich spend it, or put it in bank into their accounts or invest it into some assets they will already be part of the $M$ and redistribution, even redistribution of money, will not expand $M$.
Redistribution might decrease output due to the fact that it distorts incentives, however at the same time there are arguments that it might boost output as high levels of inequality can make harder for poor people to become entrepreneurs or realize their full potential in different ways. Empirical studies actually show that the relationship between economic growth and intensity of redistribution is mixed, suggesting that on net it has no impact on economic growth save for cases of extreme redistribution that would be well above what we typically see in developed nations nowadays (see Ostry, Berg, and Tsangarides 2014 and sources cited therein). When it comes to velocity redistribution can slightly increase it as some research shows that people with lower savings rate contribute to higher velocity (see Wang &amp; Ding 2005), however the effects are nowhere near strong enough so any transfer will have no net effect on welfare of recipients.  As a matter of fact it is generally agreed by the profession that over long time periods it is an increase in $M$ which leads to inflation (see pretty much any conventional textbook for example Blanchard et al Macroeconomcis an European Perspective or Mankiw Principles of Economics).
As a consequence of the above, redistribution can be more or less viewed not just as a transfer of money but as a transfer of resources and in fact this is how redistribution is treated in wast majority of the literature in public economics (see models presented in any public economics textbook). Many redistribution models will not even explicitly include money at all and this is not due to lack of oversight but deliberate simplification (the same way as physicists often may assume that some part of space contains perfect vacuum even if it has some particles) reflecting the view that redistribution is not simply some money creation especially when financed by taxes (even though relative prices might change which can still affect welfare but this is accounted for in the models).
Part III: Answer to the Question in the Title:
The question in your title:

Would a fair distribution of wealth from the super rich increase the purchasing power/life quality of the average person?

is actually far more sensible but I will still make 3 changes that I think are true to the spirit of what you are actually interested in.
First I am going to change the word fair distribution to redistribution to get rid of the word 'fair'. Well what is fair or fair distribution? That's a question that human race pondered ever since civilization emerged from the fertile crescent and perhaps even in prehistory without any generally agreed upon answer and discussion of fairness belongs to moral philosophy not economics.
Second I will change wealth to wealth/income. The reason for this is that I feel that like many other non-economist you do not distinguish between wealth and income and treat them jointly. For example, who is in your opinion a richer person? A old retiree whose net assets are worth $\\\$1,000,000$ because his house happened to be in gentrified area and went up in value but with measly salary of $\\\$500$ working as shop greeter, or a superstar which might have no net assets (i.e. living in fancy hotel instead of owning house) but with monthly paycheck of $\\\$70,000$? The retired person has more wealth but I think many would actually consider person in that situation poor whereas the second person has no wealth but I think most reasonable people would consider such person rich.
Third I will replace average person with low income people. The reason for this is that in any right skewed income distribution (an income distributions are world are right skewed) an average person will actually be better off than more than half of the population and optimal redistribution systems are designed to help people at the bottom not people who are better off than most (even though unfortunately in practice governments often engage in such 'perverse' redistribution as well).
Hence I will try to answer the following question:
Would a redistribution of wealth/income from the super rich increase the purchasing power/life quality of the low income people?
the answer here when it comes to income is resounding yes, when it comes to wealth the answer is not clear or rather the answer would be maybe.
Redistribution via Income Taxes
The literature on optimal income tax shows that optimal top marginal tax rates for the richest can be somewhere in rage of $50-75\%$ in the US (see Saez 2001) an generally similar estimates are also found for other developed countries.
The simulations for these tax rates are already made with an explicit goal of maximizing actual welfare of the recipients and society overall  - not to maximize any monetary transfer per se - but to maximize their underlying utility. They are based on the following optimal non-linear tax function which comes from the seminal works of Mirrlees (1971) - who in fact got Nobel Prize in Economics for this contribution, Diamond (1998) and Saez (2001).
Furthermore, the optimal tax rate even takes into account all the labor supply response and other factors in fact the formula is given by:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$ which depends on both utility of consumers and the societal utility function,  and finally $\frac{1-F(z_n)}{z_nf(z_n0)}$ is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.
If redistribution would not be able to improve welfare in poor this formula would give you zero marginal tax rates. Since simulations based on real world parameters (Saez 2001) show that top marginal tax rates could be as high as almost $80\%$ clearly redistribution can improve the welfare of poor.
An important caveat is that this optimal taxing formula does not take into account general equilibrium effects and general equilibrium effects generally (no pun intended) result in lower taxes but no reasonable estimate of the magnitude of general equilibrium effects would push the top marginal tax rates to zero. Even if we would say that in general equilibrium these taxes would be half of what partial equilibrium analysis suggests they would still result in redistribution that would be able to significantly help the low income individuals.
Redistribution via Wealth Taxes
When it comes to wealth taxes the literature is much smaller as studying wealth taxes is exponentially more difficult than income taxes due to the fact that they are rare and we also don't have good data. As a consequence the discussion on wealth taxes to some extent takes a form of twitter fights between various economists (I am referring to Summers-Saez &amp; Zucman infamous twitter battle) rather than in some proper research.  This being said some academic work was already done on this topic but it remains highly inconclusive. I will try to present both views fairly starting with the no wealth tax view and then presenting the pro wealth tax view.
Anti wealth tax view:
Some economists argue that optimal wealth tax is simply $0$. If this is so then obviously wealth tax cannot help improve welfare of the poor. There are several reasons why this would be so which are all summed nicely in this article from Larry Summers. Here is the digest:

Wealth taxation also raises practical concerns — for example, issues of valuation: Is a partnership in a law firm wealth? How will illiquid assets — like football teams or newspapers — be valued? And issues of liquidity: If someone owns 1 percent of Uber — still a private company — she will owe roughly $20 million in taxes each year, but it’s unclear where she can get this money. She can’t sell shares and, if involved with the operation of the company, is likely to be barred from borrowing against the value of her stock.


There are also family unit issues: If a couple files separately or gets divorced, do they get two $50 million exemptions? And issues of gaming: There will be incentives to use legal structures to relinquish direct ownership of assets while maintaining control of them. For example, owning assets in a trust or a nonprofit to benefit from wealth while avoiding tax liability. Granting that capital income should be taxed more heavily than it now is, and that unrealized capital gains going untaxed is a serious problem, there is also a question of just how a punitive tax is appropriate.


It is important not to be misled by the 2 percent annual rate: A 50-year old who has accumulated a substantial fortune can expect to pay more than half of it in taxes before she dies.


Imagine that a wealthy person invests in 10-year treasury bonds, with a 2.4 percent return. The wealth tax would extract 2 of the 2.4 percent return. Combined with income taxes levied at a 40 percent rate, the wealth tax could make the effective tax rate on capital income well over 100 percent. And then at the end of life would come the estate tax. While we are not aware of formal estimates of the loss in economic efficiency from wealth taxes, we suspect that if levied without concomitant reductions in income tax rates or estate tax rates, the ratio of burden on the economy to revenue raised would be far higher than with the base-broadening measures we advocate.

....

The Organization for Economic Cooperation and Development recently assessed wealth taxation and concluded that “from both an efficiency and equity perspective, there are limited arguments for having a net wealth tax.” Of the three countries with a wealth tax, two — Norway and Spain — raise an average of 0.305 percent of GDP. These taxes generate less than one-third of what the wealth tax estimates despite having a much broader base: While precise data are hard to come by, we suspect that less than 10 percent of this revenue — or 0.03 percent of GDP — comes from those in the top 0.1 percent of the wealth distribution.

Pro wealth tax view
However, Saez and Zucman the main proponents of wealth tax  would argue that the worries mentioned above are overstated (see here).
According to Saez and Zucman the main reason why the European wealth taxes failed is that Europe tolerates (and arguably even encourages) tax competition. In this article they argue that:

The specific form of wealth taxation applied in a number of European countries had three main weaknesses. First, European
countries were exposed to tax competition and tax evasion through offshore accounts, in a
context where until recently there was no cross-border information sharing. Second, European
wealth taxes had low exemption thresholds, creating liquidity problems for some moderately
wealthy taxpayers with few liquid assets and limited cash incomes. Third, European wealth
taxes, many of which had been designed in the early 20th century, had not been modernized,
perhaps reflecting ideological and political opposition to wealth taxation in recent decades.
These wealth taxes relied on self-assessments rather than systematic information reporting.
These three weaknesses led to reforms that gradually undermined the integrity of the wealth
tax: the exemption of some asset classes such as business assets, a preferential treatment of
others such as real estate, or a repeal of wealth taxation altogether.
A modern wealth tax can overcome these three weaknesses. First, offshore tax evasion can
be fought more effectively today than in the past, thanks to recent breakthrough in cross-border
information exchange, and wealth taxes could be applied to expatriates (for at least some years),
mitigating concerns about tax competition. The United States, moreoever, has a citizenshipbased tax system, making it much less vulnerable than other countries to mobility threats.
Second, a comprehensive wealth tax base with a high exemption threshold and no preferential
treatment for any asset classes can dramatically reduce avoidance possibilities. Third, leveraging
modern information technology, it is possible for tax authorities to collect data on the marketvalue of most forms of household wealth and use this information to pre-populate wealth tax
returns, reducing evasion possibilities to a minimum. We also discuss how missing market values
could be obtained by creating markets. In brief, the specific way in which wealth was taxed in a
number of European countries is not the only possible way and it is possible to do much better
today

This being said even the most extreme serious wealth tax proposals are somewhere in ballpark of $5\%$ - this is much less then the top marginal taxes that are optimal for income redistribution. It can still be argued to be somewhat well-being enhancing but clearly it is debatable of how significant would this effect be.
Nonetheless as correctly pointed by Brian in his +1 answer/comment the argument can be made that wealth tax will help to erode the political power of the rich which will ultimately help the poor. There are also counterarguments, in fact Summers argues that wealth tax might actually increase the political power of the rich but this debate is something that is outside the field of economics so feel free to read their papers and articles I linked here in full to make up your mind about this. I personally am agnostic about this power argument as there is not much research done on this issue. I suspect that even Summers and Saez and Zucman would agree that on both sides of this argument evidence is mixed.
Hence when it comes to wealth taxes it is hard to say if they can improve the lives of low income people significantly, but this is not because of a high inflationary effect redistribution via these taxes would have but rather due to the fact that optimal wealth taxes are either zero or low.
The only exception of the above is land or similar type of property which can still be counted as wealth and which is actually easy to be taxed even at high rates without generating some of the above issues (although the liquidity and difficulty of valuation argument would remain) . However, this being said most of the wealth around the world does not really consist of land so this is more or less side note.
PS: The above is just a tip of an iceberg. I did not discuss how factoring in the effect on incentives for education change the result, what is the role of capital, inheritance, consumption taxes in all of this, I also did not discussed in great lengths general equilibrium analysis, or fully factored in all dynamic effects, but all I have written still holds in spirit even if the issues mentioned here could change the actual numbers presented. If your intent is to actually learn about the field then in addition to sources already presented here and sources cited therein I wholeheartedly recommend reading the Mirrlees review which is probably the best literature review on optimal taxation there is and also the book by Inequality Atkinson who before his death was one of the top experts in the area of inequality and redistribution.