People are getting too caught up in the 250k limit.
Most bank failures result in all depositors being made whole - even over the limit. Normally by regulators “hinting” that a larger bank should acquire the failing banks assets whole.
FDIC exists to create faith in the banking system - no to just enforce a rule book on how deposit insurance is structured. The goal is for FDIC to never have to pay out in the first place - FDIC’s existence prevents the bank run from starting.
The obvious conclusion to the 250k limit (splitting deposits between banks) doesn’t really do anything to actually reduce risk in the banking system, or risk to FDIC. Splitting deposits like this is artificial behaviour that doesn’t have any real benefits to overall stability of the economy.