If just the central locking stopped, I wouldn't think the GEM is to blame.
BUT, the newer cars are CAN OBD. With the proper scan tool the repair shop can talk to individual sensors/switches in the car to find nearly exactly where the problem lies. So depending on the level of diagnosis the shop did, they may be correct.
I simply find it odd that only 1 function of the GEM is having problems. That to me says there's a sensor sending corrupt data ... or even simpler, a damaged wire or fuse issue.
I think it may be a good idea to get a second trusted shop to look at the car and see if their diagnosis differs or confirms the first. An extra 80-100 spent to possibly save 500 is still saving money.
OH, you're not playing dumb ...