Arguably, giving up your email login is worse than giving up your banking login, since attackers could use your email to reset the password on other accounts, p

Arguably, giving up your email login is worse than giving up your banking login, since attackers could use your email to reset the password on other accounts, possibly including bank accounts. For banking, a better system would allow you to generate some kind of token in your banking site which would allow the kinds of permissions you want to grant to a third party, and which you could unilaterally revoke at any time.