Don't allow password recovery.
That is absolutely not a solution. That's braindead idiocy at best. The result is that people will use one password for everything and probably write it down in a few places because if they forget it, they're fucked. Yes, people do that anyway but not allowing a password reset makes the situation much worse.
If your problem is with that "one key system", then perhaps you need to secure that "one key system" better. Twofactor auth on email hardens that single point and makes it very difficult to compromise. If an attacker is still able to compromise it, then I'd wager they'd be able to compromise those other systems anyway.