The problems in banking are caused by the government

If the reason for the existence and value of fiat currencies is not taxation then the reason (for its value) is none other than arbitrary fashion. If that is the case then the only reason to value (unbacked) bank credit is not taxation but knowledge of the ability of the government to make good on all promises, by printing more money.

Whether the reason for the value of fiat is taxation or simply fashion, it is still true that the government guarantee of bank deposits is meaningful in either case. It is for no other reason than the government guarantee that the bank deposits are valued, not because the banks are trusted, but because the government is trusted. Bank deposits are valuable or the government have lied. If the government did not want to make bank deposits valuable, there is no method by which the banks could make them valuable, they are too far into a bankruptcy situation to recover their losses. Without the government the banks would be bankrupt, it is only government support which keeps them from failing altogether. Without the government there would be no serious problem with banking.