First, you do have to remove the complete dash to get at the evaporator. When I say dash, I do mean the whole thing including the dash brace.
But, I don't think the shop was right. My daughter has an 03 Corolla that had basically the same problem, except when hers quit, it did not work intermittently, it just stopped altogether. I looked into opening up the evaporator cabinet until I checked with the FSM (factory service manual) and found out how much was involved.
I decided to put this off till last. What I found was a defective clutch relay. This is easy to check when the AC is not working. First confirm that the compressor is not spinning.
Then remove the cover for the relay/fuse box under the hood, located just behind the battery. The relay closest to the front of the car is the relay in question. The spade pattern is (front of car) =II. With the blower fan on in any position, jump the II terminals and see if the compressor engages. Remove the jumper and replace the relay to confirm it is not working.
You can also use an ohm meter across the = terminals. A reading of 175 ohms is a bad relay. You are only reading the resistor that protects the coil from the inductive kick when the power is removed from the coil.
The Toyota price for the relay is $105. The price at Advance or O'Rielly's is $15. AutoZone doesn't carry it except for special order.