That sounds a lot like "This company is too big to fail."
Well OK but I think you're leaning towards complete economic collapse, banks runs as they happened before the FDIC cannot occur anymore by default.
If you want to argue that the FDIC is in the "too big to fail" category - I'd would counter that if that really ever came to be, we'd have much bigger problems on our hands than not being able to take our money out of the ATM - we'd be talking the complete collapse of the federal government.