Many email service providers give you the option to fetch all emails from a different service not just as a one-time thing, but ongoing. I'm not sure how that could be set up when running your own email server, but I bet there's a way. Even if there isn't, you can set up automatic forwarding in Gmail.

There's a chance forwarding is better than fetching. I once had a Gmail account stolen, and account recovery was locked for some reason, but email forwarding had been set up and I was still able to get all emails the address received.

In case it's relevant, I happen to use Fastmail now and their "mail fetch" feature involves imap.