Would this alternative to fractional reserve banking work? While reading the answer to Implications of abolishing Fractional Reserve Banking on mortgages and interest rates, I started to wonder if there wasn't a simple solution to the problems that would arise with the introduction of a constant money supply.

Right now, banks create the illusion of more money existing by accepting deposits and holding onto only a fraction of them while lending out the rest.  Alternatively, could they not just package up loans into investments and essentially sell shares in the investments to average people.  This would essentially have the same effect, your cash would be tied up in the bank and you would receive some amount of interest on it.  The investments could even be structured and insured in a similar way to the current system in order to maintain some amount of liquidity.  The only big change is that people would be very much aware of where there money is when they put it in the bank, which to me would only increase the robustness of the system and reduce the possibility of bank runs, as there would be a general understanding that some of your money is loaned out at any given time.

Has this idea been studied, and are there any known or obvious flaws that I am not seeing?
How, in practise, would that be different from what is happening now? ..or what you, as an "investor" could already do?

No one is forcing you to deposit all your liquid assets into a bank account - most banks offer some sort of investment-opportunities, where you can buy into different funds with different risk-profiles.

Next thing; suppose banks take all deposits and lend out to different customers for various purposes and they receive interest on those lendings - some of those interest are transfered back to you for allowing the bank to use your money. This would actually be the same, as putting your money into a fund. The bank "invests" your money with other customers and recieve some sort of yield on that investment (interest). Difference will be the risk-profile. By "investing" in the bank you recieve an extremely diversified portfolio, funding everything from consumer goods, cars, houses to actually backing large corporations. Most of your investment is being placed high up in the hierarchy of priotization if the lenders go into default, which makes your investment "safer" than if you go and buy bonds in that same company (but obviously also recieves a lower yield).

Your proposal will put peoples money in a different place in the hierarchy of debt if customer/lenders go bankrupt, actually increasing the general risk-profile of the bank.

In all everything comes down to your risk profile - which exposure do you want your investment to have. The risk of a AAA rated sovereign bond yielding -0.5%, receive 0% on your deposits in a government backed bank, 5% on a BB- mutual fund or 25% on an emerging markets bonds?