In reality, there is no such thing as a "bank" in the sense that it's a big vault of money so you don't lose it in a house fire.
Every bank today is really a Highly-Levered Bond Fund. If we increase FDIC insurance to 100%, without commensurate changes in what banks are allowed to do with customer deposits, this seems like it would create a moral hazard where banks would be allowed to do all kinds of crazy shenanigans because they know they'll be bailed out.
To be clear, I'm for 100% FDIC insurance, but if we get this without getting much more stringent duration risk transformation standards (e.g. don't load up on long-dated treasuries in a low-rate environment), I think it could actually make the banking stability issue worse :(