Would this alternative to fractional reserve banking work?
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?