First, you get a backup of all the old emails (person in the case lost access to 15 years worth of emails). Second, the sync is configured, so you have a backup of freshly arriving emails. That's reduces your risk already but you don't stop, once all of the above is setup you setup a filter "to: [email protected]" and move those to a separate folder and then you just act on every email that you receive there. If it's from a robot, you go to that service and update your email. If it's from a person, you ask them to change your email in their contacts and so on. In fairly short amount of time, you have the issue resolved completely.
The funny thing is that once migration is done, you can configure Gmail as POP3 client to your email server and enjoy Google's UI while it lasts with zero risks. Yes, it'll be able to use SMTP server as well.