It's justified via statistical multiplexing, just like when an ISP sells a total of 1 Gbps of bandwidth but only has a 256 Mbps upstream connection.
So long as only a low percentage of demand depositors ask for their money back at any one time it's not a problem. But when everyone decides to withdraw their money at the same time you have a liquidity crisis. Liquidity problems aren't so bad anymore though -- the Fed steps in an lends all the cash you need against your long term assets. The real problem is when those assets go bad. Now you don't have a liquidity problem you have a solvency problem. The only thing the Fed can do at that point is to simply give the bank money to make up for their bad investments. Which is exactly what they did and are doing, albeit in an obfuscated manner.
Great stuff. Although I should say that there's more you can do with bad assets than just give people money. My understanding is that classic resolution is where you create a "bad bank" [1], a holding company for all the bad assets. You then fire the bank managers, put less dumb people in charge, recapitalize the banks, and use the bad bank to slowly realize the value from the assets, which are often not totally bad, just part of a cyclical slump. And you also update regulations and capital controls so as to reduce the chance of a similar mess next time.
A lot of people are critical that after the 2008 crisis very few people got fired for FUBARing the world economy. I get why the US ended up doing that; people were scared of anything that looked like more instability. But I think it was a mistake.
The Fed can't do something like that. It usually takes the involvement of a bankruptcy court, though the FDIC has some powers it can exercise independently. A full Swedish style resolution would almost have certainly required new legislation.
All that said, I agree with your underlying point that just giving insolvent banks money wasn't a great solution at all, though it was minimally sufficient to prevent bank runs at least on formal banks. There were some runs on shadow banking institutions, though in at least one case -- money markets -- the government stopped one by guaranteeing them as though they had been insured banks. That too was a mistake in my opinion.
So long as only a low percentage of demand depositors ask for their money back at any one time it's not a problem. But when everyone decides to withdraw their money at the same time you have a liquidity crisis. Liquidity problems aren't so bad anymore though -- the Fed steps in an lends all the cash you need against your long term assets. The real problem is when those assets go bad. Now you don't have a liquidity problem you have a solvency problem. The only thing the Fed can do at that point is to simply give the bank money to make up for their bad investments. Which is exactly what they did and are doing, albeit in an obfuscated manner.