Why doesn't teh mechanic just take out the Climate controls to troubleshoot it?? oh, that's right, he just sounds like he wants to take all your money.
Just pull out the stereo and then undo the two screws and unhook the temperature control cable so you can pull out the climate control yourself and repair said broken wire.
#20 & #21 are near where the screws are that you remove to slide the unit out once you have the stereo removed.