An account at a bank is a liability of the bank. The bank is liable to present the funds in the account.

To be solvent, the bank’s books have to balance, meaning assets to offset liabilities. When a bank makes a loan, that’s an asset of the bank since the loan yields interest. The bank creates the loan account balance (asset) and the deposits for the demand account (liability) at the same time.

So in effect, the bank does have infinite fiat, it just only comes into being incrementally when the bank uses it to purchase assets (issue loans).