But this means that all users of USD are providing an insurance pool for the banks for free. Doesn't quite seem fair.
There are some decentralized stablecoins (stablecoins backed by crypto assets + market rules that ensure they keep the peg) that have explicit insurance pools + rewards for being part of that insurance pool. For example, Liquity [0] allows users to deposit the stablecoin and earn rewards + favorable liquidation positions (a 10% discount) in return for being the first layer that purchases liquidations.
Not exactly sure what the equivalent would be in the traditional finance world, but I could imagine some sort of higher interest savings account that isn't FDIC insured, and takes losses first.