It doesn't feel like the heater is on, it feels like the ambient temp. The air conditioner won't kick in because it thinks the temp is 60 degrees which is what the display says even though in reality it is about 20 degrees higher.
Why do you say this?
There is no display for interior temperature! Where are you getting that it thinks it is 60 degrees inside? Do you have a scan tool hooked up and looking at the sensors? The climate control is not like the thermostat in your house. It never displays the actual temperature. It only displays the temperatures you set it to.
Also, no matter what it thinks the interior temperature is and the exterior temperature is, it always turns the AC compressor on. It just adds heat from the heater to get to the right temperature.
If your compressor is not running, then it's because of one or more of these conditions:
1. The system pressure is too low
2. The system pressure is too high.
3. There is a wiring fault.
4. There is a DATC fault.
5. There is an engine problem.
6. The evaporator is below freezing (or its temperature sensor is faulty).
None of these things show up on the display.
One more time,
The 60 degrees that you see on the display is the temperature you have set the control to, not the temperature that it thinks the car is. If there are two temperatures up on the display, then you have it in dual mode, and at least one side is set to cool to 60 degrees.