Banks hit with runs. Banks don’t keep a lot of cash on hand. When they are hit with mass withdrawals in a panic they have to have a way to hypothecate their illiquid assets or an otherwise sound bank can be bankrupted. This happened on a huge scale from 1930-33 when the Fed failed to act as lender of last resort and let nature take its course. One third of American banks simply vanished, taking their depositors assets with them. No FDIC.
If just one bank is having problems they can sell assets to other banks. If the public at large panics then that won’t work because too many banks get hit at the same time and they can’t bail each other out. Then you get a system failure and a replay of the 1930s.
When the housing bubble blew banks found themselves holding a trillion dollars of worthless paper. I suspect that a lot of major banks that we all know may have been technically bankrupt or close to it. The Bernanke Fed was aware of the major blunders of the 1930s Fed and their policy was to provide as much liquidity as possible to keep a cascading failure from starting.
Did you read that I posted the link to? Tamny has an incredible answer about insuring banks. I really would like to suggest you read the rather short book, 180 pages, and then let’s have this discussion. We’re asking and addressing the wrong questions.