Long story short, you’re caught up in monetary policy. To mitigate these risks, you can use deposit accounts that sweep across multiple banks to get the cash insurance you need, and your finance/treasury department can invest the rest in short dated treasuries (which are backed by the Federal Reserve with no limit, and are considered risk free) or money market funds which contain only these same securities.
https://en.wikipedia.org/wiki/Certificate_of_Deposit_Account...
https://www.chicagobooth.edu/review/safest-bank-fed-wont-san...
Maybe the fraction should be made larger than 0%? Or maybe bailouts should come with more strings attached including actual penalties for executives and board members?
No, the banks have to pay premiums to FDIC for the insured amount, period. They have to pay premiums for these first $250k that get insured; it's "automatically insured" because they don't get a choice, but that insurance costs a market-price premium out of which all the potential payouts come.