Milton Friedman called the creation of FDIC the most important legislative innovation of the Depression.
The FDIC makes an unsound institution look sound. When banks pyramid with a 10% reserve requirement, there is no institution that can bail them out in the case of a massive bank run, except the printing press, and that will completely destroy the value of the currency.
The FDIC is only required to have something like 1.85% (maybe even lower) of total deposits backed. If there were a serious crisis and people withdrew their money thinking it was safe, there would be panic when the FDIC ran out of money. Currently I think the percentage has fallen below 1%. There’s nothing the FDIC can do in the event of a major bank run.
For the record, the FDIC’s current insurance fund ratio is 0.40% of all insured deposits, and the minimum is 1.15% as mandated by Congress.