You said "I picked it up just before they closed today. When I stopped for gas on the way home I saw something hanging down from the car. When I popped the hood I saw that the AC belt was broken. When I looked closer I found that the clutch on the compressor was siezed up. I can't figure out how a dealership tech could have missed such an obvious problem."
Very easily, they created the problem.
Happened to me when I got an oil change at a BMW dealership, the next day I had my rad flushed at a local garage.
The Heat shield was riped down along three bolts, I noticed a rattle after having the oil changed and looked under the car but never noticed the heat shield hanging low.
The next day when the car was up on a hoist I looked, obviously if the car was up when they did the oil change they would have seen the heat shield hanging.
I figured out after they riped it down themselves to create work.
Be careful when you leave your car with the stealership, it is not a question of trying to make extra money by scaming customers, it is a question of survival for some dealerships.
If they don't they will be out of business!