>they will get serviced BY Rolex.

Gray market sales exist and Rolex will not service anything bought through the gray market, rather than directly from an AD. This leads to a lot of non-Rolex watchmakers who can work on Rolex. If they also do some check to see if they happened to be registered or stolen, I’m not sure.

I think it would be a bigger issue for less popular watches with more complicated movements, where the local watchmaker would be in over their head.