It's not a theoretical problem, but there's a practical one: Banks themselves loan out the money, so a run on a stablecoin can cause a run on actual banks. And FDIC isn't enough to cover the full amount.
This can be mitigated by using many banks, but it's hard to find banks willing to deal with cryptocurrency. So in practice all the deposits are at 1-2 banks.
A stack of paper USD in a safe would work, but there's risk of theft/fire/etc.