A bimetallic strip thermostat with a ‘cool/off/heat’ mode switch and ‘on/off/auto’ for fan control is all anyone actually needs with a fixed speed compressor and generic furnace with a fixed speed fan.

If you have a fancy modulating furnace and variable speed compressor or a variable speed heat pump, a thermostat that uses a PID would work better.

Fixed speed compressors are okay for AC condensers but for tighter applications like heat pumps a variable speed drive offers a good efficiency gain, the problem is there is no standardization established for the driving of say a variable motor since they are all redesigned and unique so when the controller goes kaputz you are out 3-4000 dollars for something that costs 30 dollars to manufacture