To begin with the aux fans are not turned on for a/c purposes by a temperature sensor. They are turned on by a high pressure switch. They are not turned on for the purpose of making the a/c perform better. They are turned on to bring down the high side pressure to prevent system damage.
It sounds as if the refrigeration portion of your air conditioner is working correctly. The cabin temperature is regulated by the introduction of heat. This is done by regulating the heater valve near the battery.
The pushbutton control unit supplies a signal to that heater valve to adjust cabin temperature. If the pushbutton unit sends no signal to the heater valve it is open allowing heated coolant to circulate through the heater core.
That said, don't rush out and buy a heater valve or pushbutton unit without doing some investigation. There are a number of elements involved in the overall ATC system. There are temperature sensors, a fan to draw sample air over the temp sensor, there is a measurement sensor to measure air across the evaporator, the valve, the pushbutton unit and associated wiring.
I think I would start by somehow plugging the coolant line leading to the heater valve. After plugged, drive the car to see if you then get cooling. If so, then it is time to go down this road to determine what is causing the heater valve to open.
Once you determine that this is your problem provide smoke at the little grill near the rear view mirror with the ignition on. The smoke should be drawn into the grill. If not, then find and check the little fan, it will probably be located behind the glovebox.
With those two simple things out of the way, see if there is a signal to the heater valve. You will need to research and see what to expect. I think it is a duty cycle type signal, but I expect that a voltmeter will show up SOMETHING if it is giving a signal. If it is not, the problem is likely to be the pushbutton unit, but investigate as best you can before shotgun troubleshooting and replacing it as an experiment. Also before replacing it, take the back off and heat up the solder joints. There will sometimes be a cold solder joint that is the problem.
Also when replacing the pushbutton control unit check aux water pump current draw. If it is 1.3 Amps or more, replace it, if you don't you will knock out the new PBU.