For those interested in this, I strongly recommend Sheila Bair's "Bull by the Horns". She was the head of the FDIC up to and during the 2008 financial crisis, and this is her memoir. She's a fiscally conservative Republican, but one who strongly believes in the value of regulation as a way to create a sound economy so that all citizens can thrive. The book was fun to read, and it gave me a much better understanding of the forces at play and why good regulation of banks is immensely valuable to us all.
Borrow short and lend long - great work if you can get it. Of course, this practice is fundamentally unsound (it provides nasty game-theoretic incentives to participants), but it tends to work "well enough" in practice that no one really cares, especially when there is a lender of last resort who is able to print money at will.
which states that "The fund had a balance of negative $7.4 billion as of Dec. 31, though that was an improvement from the $20 billion hole it was in at the end of 2009."
If it is simply a classic bank run, and everyone wants their money back now, the lender of last resort could take over all of the distressed bank's illiquid assets and, over time, recover some or all of the lent moneys.
In this case, what the lender of last resort is really doing is making all illiquid assets liquid.
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.
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.
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.
But each bank loan is balanced, to the reserve ratio that bank operates at, to a corresponding deposit.