For example, setting an email forwarder to an account an attacker controls in most cases won't even be noticed. I think it opens more attack vectors than the good it could do to have this kind of integration rather than just a password manager.
Giving more control to a single manager (in this case an email account) also means you will have to set greater security standards for it. For example, are you going to type your password (which also controls all your accounts) to your friend's, school's, airport's, etc's computer that could be infected?
Passwords are insecure? Of course they are insecure. That's why we are trying to implement two factor authentication. But having 1 account with 2 factor auth controlling 20 accounts with 1 factor auth isn't exactly helping. At all.