I mean there are a myriad of ways to avoid this:

1. Log out
2. Use container tabs or profiles in your favorite browser
3. Use dedicated apps like FreeTube, GrayJay, NewPipe, Invidious, LibreTube, etc. that don't even ask for a login.

So is there an easy way to migrate all your stuff to some other email service and what are some good ones?

So you can set up a free Proton Mail account and then connect it to your Google account and it will begin archiving all of your emails, past and future. It also has a way better web client than Gmail (not saying much).