I would like to clarify that fiat currencies and banks are slightly different things and are not quite identical, and the same banks, subject to the permission of the financial regulator, could well perform operations with cryptocurrencies.
The reason we are using bitcoin and other cryptocurrencies is to avoid a third party to hold our coins and suppose the financial regulators allow them to transfer cryptocurrencies, what is the purpose when you do not need a third party. You can start a locker system to safe guard your coins in their vault which is much safer if you are holding huge amounts.