The worst part of this analogy is that you do not have to run a mail server to not lose control of your emails and your address. You cab download the mail and switch your DNS at any time and leave a mail provider.

That's not true of a bitcoin custodian. They could rug you. You can't download / copy bitcoin or switch without the custodians permission.

There are many better analogies. Passwords. ID. Domain names. But they just suck less.