If it was be that simple. In that case I would have to go to the bank for every transaction/payment I want to initiate online. Banking app doesn't work for jailbroken devices. Using PC to access banks website works, but transactions still require 2FA and they don't support any other 2FA flow except the one in the app.
You don't have to go to the bank for every transaction, you can just go there once to close out your account and open one somewhere that doesn't require that.
Depends though what you mean by "do not use Google". Having an Android phone with a Google account logged in will not affect you much. If they would block one account you just create another.
Having all your emails on Gmail and used for external services (bank, insurances, etc) is a different story though. I prefer to pay my email provider, at least they will care a bit more than they do for a free account...
I'm surprised, most banks I've come across force sms or phone-call 2fa only. A rare few allow generic TOTP authenticators, and maybe one or two has an app as an option. And I've only come across one bank that detects and warns for root access. Is there no "jailbreak hide" on ios?
There's always a workaround. There are banks with far less annoying root checking and you can just switch. Many banks allow SMS or a physical authenticator for web banking or 3DS 2FA. There are also many was to bypass root detection. If your main problem is 3DS 2FA for online card payments, get a proxy card.