I can't tell from what you wrote, but did you actually try the A/C button all by itself, without using the auto climate control? Fan on high and a/c on?
What about when you turn the vent to defroster only. Does it click on then?
+1 on this. You have to give us more information.
When you say "the compressor doesn't come on", what does that mean? Did you push the "AC" button and if you did, did the light come on?
Here's the odd thing about the "Auto" climate control feature: If you manually turn off the AC, it will NOT come back on, no matter what you do as far as adjusting the temp. You have to manually turn the AC back on in order to get AC again.
And yes, I had to learn that one the hard way, too (I like to turn off the AC when I'm about 5 minutes from my final destination for the day so cold air doesn't sit in the AC ducts - this helps prevent mold from forming.)
If you push the AC button and the light comes on, then adjust the temp down to its lowest setting and the AC should kick in noticeably.
FWIW, I hate the HVAC controls on the 4th gen. Why take something simple and make it complicated? I've finally gotten where I understand the controls a bit better but it shouldn't be that hard, Toyota should have come up with more intuitive controls.
I also hate the fact that with the headlights on, the panel lights are so dim I often can't tell whether the AC is on or not. Does anyone know if there's a way I can put a brighter LED in there? Or maybe just a different colored one? The amber LED isn't as easy to see during the day as the green LED on my 3rd gen.