That's why you need to own your domain. When they lock you out, you setup a new mailbox elsewhere and point your domain to it. That covers new mail. And the archives are still accessible because you've been forwarding mail all along.
And, if your mail client’s UI allows, you can even send from <whatever>@yourdoma.in to avoid disclosing your “real” address when you reply.
My favorite use-case: you sign up for foo.com with foo@yourdomain.com. Then foo sells their contact database, and now you’re getting viagra spam all day. Ok, just add a filter to auto-trash anything addressed to foo@, and the problem is solved.