No, that's a little different. The problem with password reuse is that if someone breaks into a site and steals the passwords, or a malicious administrator steals your password, they can reuse it. The public/private key separation of ssh keys means that that is much less of an issue. You can't login to another server with my public key.
If the integrity of a password was dependent on a one-time unlocking mechanism secured on my local machine, then yes, you would only need one password. Passwords are not public key cryptography.