I doubt a sensor would react directly the reliably to "engine load", by which I mean when you are going uphill with passengers, the engine is working hard. A bad sensor has no knowledge of this--it just gives a wrong reading randomly, seems to me.
Given your symptoms, I think you have a blockage of some sort in the cooling system, or air in the system.
What does the garage mean by "no indication" of head problems? There's only one way to know, and that's by testing. If they haven't tested, they don't know.