Your client can choose the password for a chatmail server account arbitrarily. You would need one less thing to remember if it was generated deterministically using a password derivation function (hash) based on the private key and the hostname.
Querying the email address corresponding to a public key fingerprint is already possible using public key servers - you could submit yours to one already:
https://en.wikipedia.org/wiki/Key_server_(cryptographic)#Keyserver_examples
Yes, it would be ideal if such a backup contained everything but message content that could easily compress to a manageable size:
It can then be printed on a single sheet of paper: