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?

GPG doesn't easily lend itself to be wrapped, or even used for that matter.

You are welcome to try.

Tried to explain GPG, public/secret keys to non-tech guys. Not fun.

Something like that must be easy to use, and easy to understand to be used by the masses.