[rprospero]

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?

[NaNaN]

The keys can be printed out on a paper. If you set a strong enough passphrase for the secret keys, you can upload them somewhere, too.

I should rarely use my special keys (or passphrases of the secret keys) so as to keep them safer. And simpler authentication is more efficient for normal web services. GPG authentication may not be fast enough.