If government distributed all their bitcoins to us, and took all the money rich have, and gave it to us, and if medicine never cost a dime, and if free food dropped down from the skies that would be a great world.
However, we live in a real world and sometimes we need to know how much we can safely assume could be done. Preferring bitcoin to be insured like that is a big ask, considering the volatility of it, we are talking about a huge deal here that would ruin a governments accounts and treasury so there is no way that any sane nation would do something like this, specially when they know it wouldn't bring them that much vote to help offset the people who will hate losing money because of this since it would cost a lot.
FDIC doesn't insure the value of the USD. FDIC merely provides protection in the case of a bank failure. The USD, as denominated in BTC terms, is equally volatile. Or USD denominated in SPY. Or USD denominated in Gold. To various extents obviously.
So I don't see your point in the volatility of BTC, one BTC = one BTC. One USD = One USD. Trading pairs between USD and BTC vs other assets go up/down all the time.
In a situation where the FDIC or another federal program insures BTC deposits, obviously the bank would need to have reserve requirements. Perhaps that requirement is 90% or something extremely high. Or it's 100%.
Doesn't really matter bc it's the backing of the US government that matters most in a situation where the reserve requirements are high.