Does everyone who’s following the old account automatically refollow you when you do that?

It doesn’t port over any old comments/posts, but I’m pretty sure that when anyone @'s you, it’s forwarded to the new account.

IMO it’d still be useful to be able to use an identity you control, like a domain name.

Mastodon already does this

It’s worth pointing out that while ActivityPub doesn’t currently support account migration (although there are proposals in the works for how to do this), Mastodon does have a weak form of support right now.

You can create a new account on another mastodon instance, then you’re able to point your old account to your new account.