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Deposits are insured up to $250K, I wouldn't worry about it. When Credit Unions ran into trouble in the 1980's there was not enough money in the insurance pools to bail them all out. It's a different pool, but the point is the same. The FDIC only has enough cash to cover about 1% of deposits. Most bank which are forced to close (though not the fiasco of 2008-2009) were implemented when banks were still solvent, i.e., they had more assets than liabilities, but did not meet regulatory margins. The banks and their assets were sold, with the buyer, usually a competitor, honoring all outstanding liabilities.

Your comment also points out another problem: Moral Hazard, common to the insurance industry. Orthopedic surgeons often set up business near ski resorts. Their patients tend to be well insured. Would people be as willing to ski if skiing accidents were excluded from health insurance coverage, or required a separate and expensive rider? (The same argument could be made for motorcycling.) "Free" bank insurance relieves depositors from prudential caution about the solvency of banks, and leads depositors to look for the best insured returns.