[StickMon]

Want some of my hot air?
I have the exact opposite problem. Back in August I got stuck on full hot.
Everything else works, but I can't hear the blend flap actuator.
Not sure if actuator failed, controller failed, or temp sensor failed.
Maybe the temp sensor failed and the car thinks it's absolute zero.

But back to you:
I think water flows through the core at all times, and temp is controlled by the blend flap.
If you turn the car on, but don't start it, and stick your head under the glove box, you should be able to hear little gears whizzing when you move the controls.