Why am I not surprised by the dealer's suggestion
It will be fuel for the infinite improbability drive if all 4 door handles went out of order at once and everything else is working fine. And you replaced the driver side with a newer, less used one. If it were me i would leave the handles alone and look at the control unit. I dont remember if your CA was working when your door handle was heating up, but it may have damaged the control unit, or it may need coding after replacing the handle. If you look at the CA document I posted in the other thread you will see that the 4 door handles communicate with the CA thru 1 line, and all other antennas (that work) are using different channel. The dealer tech probably checked the actual values and saw no communication with the door handles and decided they are all bad, but if it were me i would suspect the CA module.