What would be the economic effect of widespread instances of individuals paying off consumer debt? In light of the economic impact of the recent shutdowns in response to COVID-19, the Australian Government is considering bringing forward planned tax cuts, that were do to be implemented gradually over the next five years. The tax cuts themselves aren't too large compared to other economic stimuli currently in action, but I read the following statement on a Guardian article:

The Australia Institute paper, by senior economist Matt Grudnoff, concluded that bringing forward the income tax cuts was an “ineffective stimulus”.
He cited the “simple fact of economics that high income earners are more likely to save or pay down debt with the tax cuts, rather than spend the extra funds to help stimulate the domestic economy”.
“A more effective way to stimulate the economy would be to invest heavily in direct employment programs or focus on supporting those who are doing it tough by maintaining or increasing the current rate of the jobseeker supplement.”

This left me wondering what happens to the wider economy when people decide en masse rather than spending their disposable income on consumer goods/services, to instead  pay down their debt and save/invest?
This left me wondering what happens to the wider economy when people decide en masse rather than spending their disposable income on consumer goods/services, to instead pay down their debt and save/invest?

tl;dr: Answer depends on the situation/time horizon you are talking about. In long run increasing saving and investment will have no negative impact on economic activity. In fact it can even lead to higher long-run economic growth.  However, in short-run and in recession (especially one where economy is in a liquidity trap) it would lower everyone real incomes and depress economic activity.
Full Answer:
To see what mechanisms are in play here we can use little bit of math. This can be done with the standard goods market equilibrium model (see Blanchard et all Macroeconomics: a European Perspective)  Let us start with output/GDP definition for closed economy:
$$Y =C+I+G \tag{1}$$
where $Y$ is the economic output (which in economics must also be equal to people's incomes so I will be using output/income interchangeably). $C$ is consumption, $I$ is investment and $G$ is government spending. In order to get some meaningful answer from the above identity we have to specify what the consumption is. In order to make everything simple lets assume linear consumption function:
$$C= c_0 + c_1 (Y-T) \tag{2}$$
Where $c_0$ represents your autonomous consumption -  consumption you will consume regardless of your income, $c_1$ is your marginal propensity to consume, if $c_1=0.75$ that means you will consume $3/4$ of your disposable income and save the rest and finally $Y-T$ is income minus taxes which is the disposable income.
If we substitute $C$ back to the GDP definition and solve for $Y$, we get the goods market equilibrium:
$$Y = \frac{1}{1-c_1}\left( c_0+I+G-c_1T\right) \tag{3} $$
This shows that the level of economic output $Y$ will actually depend both on consumption and investment (and investment in turn depends on private savings as it is the sum of private and public saving). However, it also shows that if people's marginal propensity to consume increases (that is when $c_1$ becomes higher) the whole output multiplier $ \frac{1}{1-c_1}$ becomes higher and hence for any level of autonomous spending, investment or government spending the output will be higher.
Nonetheless the above is all just a short-run partial equilibrium analysis. Such analysis is appropriate when we are in recession and where investment is unresponsive because for example an economy is in a liquidity trap (a situation where increase in savings wont necessary fuel more investment). However, in long-run we have to acknowledge that investment is not just independent as the formula (3) suggests but also depends on income and interest rates as investment as mentioned above arises from private and public savings.
If we make just one very simple change to the model above and assume that investment is given as $I= I_0 + d_1 Y -d_2 i$ , where $I_0$ is autonomous investment, $d_1$ is the fraction of income that is invested, $d_2$ is parameter which determines how investment respond to interest rates and $i$ is an interest rate, then the goods market equilibrium will be given by:
$$  Y = \frac{1}{1-c_1-d_1}[c_0 + I_0 + G − c_1T] - \frac{d_2}{1-c_1-d_1}i \tag{4}$$
In this case you can see that the multiplier is given as $\frac{1}{1-c_1-d_1}$ so multiplier is given by spending not just on goods and services but also by investment spending - if people lower their marginal propensity to consume but increase marginal propensity to invest to offset that nothing will change. Furthermore, ceteris paribus increasing the amount of savings lowers the interest rate (which is the price for savings that can be used as investment).  So once we let investment to vary depending on interest rates and income we see that what matters is the amount of all spending no matter whether it is spending on goods and services or investing.
In addition the above is still missing explicit model for money-market and behavior of firms if I would add that in a result would show that in the long run all savings are transformed into investment either directly (which is modeled here) or indirectly through their effect on overall price level but an exposition of full model would be too complex so if you are interested you have a look at some undergraduate textbook (all models presented here are taken from the above mentioned Blanchard et all Macroeconomics but they will also appear in any standard macro book with perhaps different notation).
Furthermore, the discussion above does not factor in economic growth and saving and investment is crucial for economic growth. The main growth model used in modern economic literature are the Solow growth model in which increase in saving rate can result in short to medium term higher economic growth, but not really in  a long run per capita growth which will in the model be determined by the rate at which technology progresses. However,  for that to happen the economy still requires some level of saving - the model just shows that trying to increase the level of saving in the long run steady state wont make difference (See Romer Advanced Macroeconomics).
However, recently in literature endogenous growth models are becoming more popular (Romer even got his Nobel Prize in economics in  2018 for his work on them) and in such models savings rate actually can increase even long run economic growth (again see Romer Advanced Macroeconomics).

PS: This is little bit of an tangent and hence I included it as a post scriptum but actually this:

He cited the “simple fact of economics that high income earners are more likely to save or pay down debt with the tax cuts, rather than spend the extra funds to help stimulate the domestic economy”.

this is actually not entirely accurate - or at least taken out of context (probably journalist misunderstood statement about particular tax cut as a general one). You can always design a tax cut that specifically targets poor. For example suppose we have flat tax rate of $40\%$ we can always decide to cut tax rate to let's say $10\%$ only for people below some income threshold. Rather the issue is that taxes generally have lower multiplier attached to them and hence they affect output differently. From equation (3) you can see that if government spending $G$ increases by 1 the output will increase by $\frac{1}{1-c_1}$ but if the government decreases taxes output increases only by $\frac{c_1}{1-c_1}$ and on interval $0&lt;c_1&lt;1$ the former will be always larger as on that interval $\frac{1}{1-c_1}&gt;\frac{c_1}{1-c_1}$. This also holds just in short-run though.