That and you can't migrate your message history.

The latter isn't fundamental to the design, since your new server could import the posts and old server could 301 permanent redirect each historical post. Though you'd have to redirect every piece of the thread which seems like a hard problem.

But the former is fundamental, at least as long as ActivityPub uses internet domains as the root of identifiers. Whoever controls the domain controls you. Even having your own domain and hosting your own instance doesn't fundamentally fix that, since your registrar controls it, but it helps.