Here an off the wall sort of idea.
If the only symptom of overheating is indicated by the temp gauge the car may well not be actually over heating.
The gauge is electric which relies of the temp sender to give a variable voltage signal to the gauge by varying the electrical resistance according to coolant temperature.
Now lets consider what circumstances that could cause a false reading when the heater is set to cold.
As the heater valve is cable operated, maybe it's actually acting as an ground cable depending on it's position.
This could be due to some damage to the insulation of the wiring rubbing against the cable or perhaps a ground wire has a dirty or bad connection causing the current to earth via the heater cable.
When the obvious causes aren't the culprit, the one has to look to less obvious and obscure causes.
Can you beg, borrow, buy, bludge or steal an inferred heat temperature reading gun? Use this to get a better idea as to if the car is actually overheating, by pointing it at the radiator, taking a reading and comparing it to the gauge reading.
Most OEM gauges fitted to cars are rubbish built to a price, they're not calibrated scientific instruments, and even if the were calibrated properly at the factory they would still need to be periodically recalibrated anyway.
Clutching at straws here............