You can use imapsync to keep a local synced copy your email in maildir format. That, along with a domain you own, solves the "I own my own email sycned to my server" problem pretty much completely. Switching email providers is just syncing to the new server (which imapsync supports) and updating your MX records on your domain.

Some email clients (like aerc) can just operate on the maildir directly. That's what I do, but there are lots of ways to do this.

One option is to just keep using your email client the way you are now talking directly to your mailserver, and just rely on the imapsync side to be your owned backup/migration path.

Alternatively you can run your own local IMAP or POP server like Dovecot backed by the local maildir. So that's imapsync -> dovecot -> thunderbird (or similar). It's a little more software to set up, but once it's set up it should be very reliable and gives you total control over what email client you use and full ownership of your data.