You can setup a rule in Gmail, and tag them when they get to your new inbox. Use Gmail to add a junk CC, or something if the new provider doesn't have strong automation capabilities.

Then use the tags to go back and change the email on file (if necessary).

After a year if they haven't contacted you then they obviously aren't that important.

I found a lot of emails but I didn't find a lot of emails I cared about enough to update when I did this.

That's a smart way of going about it. I threw caution to the wind because I stopped using my Google account for services years ago. We'll see how that affects things in the future.

Gmail might have an autoforward thing