They could have their own bank but that’s not the goal of stablecoins. When stablecoins were first released it was so users could transfer money from exchanges such as Bitfinex to exchanges which had tons of altcoins which the bigger exchanges didn’t have.

So the user wouldn’t need to sell their US dollars, buy bitcoin, send bitcoin to some alt exchange and then sell that bitcoin for US dollar and finally buy what alt coin they want. This is too many transactions and hence why stablecoins were introduced because you could send back and forth between exchanges.

There is no point is having a bank for stablecoins however. Just send to your exchange, sell for fiat and withdraw that fiat to your bank and buy whatever you want.