So turn signals do not work but four way hazard flashers do work?
And a new multi-function ("Stalk") switch did not fix this? That would lead me to this test:

Place multi-function lever in LEFT or RIGHT signal position.
If they don't work, slowly press in on the hazard switch. Wiggle it. Jiggle it. See if moving the switch gives you a spot on the travel of the switch that enables turn signals. This would indicate that the normally-closed section of the hazard switch that feeds 12 volts from the ICM output is bad. That means you did not need a multi-function switch at all. (Power for hazards comes from fuse 35 thru the hazard switch. Power for turn signals comes from fuse 4 in the ICM. This power is switched by the Hazard switch's contacts.)

The actual flasher is in the Body Control Module. But its fine because the Hazards work.

I think your hazard switch is bad OR the multi-function switch was bad and they didn't get the connectors plugged in right when they put the new one in. Also, they apparently did not get the airbag squib wire connector plugged in right, either, and the ORC (occupant restraint controller) detects this and is trying to warn you.

Many owners here in the states have replaced their multi-function switch themselves and most had no problem. It is a common failure item as our cars age. I think your shop is the real problem here.