The Apple Mail app on iOS and MacOS is pretty decent. No need to use the official Fastmail client (if you are on Apple and prefer the simple Apple alternative).
Why?
You go to your register and point your domain to another email provider, and you’re back in business.
All your previous emails are lost, aren't they? Even you have local backup of them somehow, they are not equivalent to emails saved to your server?
I've done that in the past, don't recommend it unless you have a real need and know what you're doing. It wasn't hard, but just extra work that I wasn't sure was worth it.