My bet is that the ambient sensor in front of the radiator is not plugged in. I would doubt that most shops would think to check that.
I just checked in my car by unplugging it, and it definitely shuts off the compressor when unplugged, even though the AC control light is on as if the compressor was.
A way to check this without even popping the hood is to turn your key on and see if the multigauge displays the outside temperature or if it says "---". No reading = not plugged in.
After you pass that test I will shut up about everything being plugged in
