Today, I moved my hosting to a newer platform. This went surprisingly well, because I ignored all safety precautions I usually take. I simply called my new machine the same and set the DNS TTL to 300 beforehand. This way, I could simply rsync my old machine to the new one, turn off all services and run a final rsync.
Then I booted the new machine, moved DNS and done.
Usually, I would have given the new machine a new name, but this would be a hassle with TLS etc., because I currently include the hostname in the TLS Certificates for some services. Anyway, I have some cleanup to do on that front, so a future migration does not feel as hacky...