Other than passwords, client certs cannot be guessed or generated with wordlists, they can not be used to authenticate to other sites even if stolen, and they cannot be forgotten (at least I tend to forget passwords for rarely used sites way more often than I lose data through HW failure).

So I'd say there are some advantages over passwords. And most of the functionality is already there, the only thing missing is a browser function to generate a keypair for a domain and send the public key, without much user interaction.