The problem with using GPG for password retrieval is that we're assuming that the user is irresponsible enough to lose their password, but responsible enough to maintain backups of their public and private key.
To put it differently, if you have gpg authentication available, why are you even bothering with a password in the first place?
To put it differently, if you have gpg authentication available, why are you even bothering with a password in the first place?