My email accounts' passwords are far more important than that of my bank account, and there's no way I could put up with typing them in every 15 minutes.
On top of that, chances are that a bank login saved in a password database, which has 2FA and other sensible precautions, is probably kept safer for any one individual than the bank's systems themselves, what with the huge legacy cruft they suffer from. No-one would be able to walk into my computer, or call it, with some faked documents and social engineer themselves into my password database.