Brendan,
I did a min-search on your engines operating temperature...the general consensus was 160F.
I am with you, I think 180 is too hot for this old stump puller. When you checked the radiator with your thermometer, where was the needle in the dash temp gauge. I'll bet if the gauge works, the needle was higher than mid scale, or normal.
If you can't control the engine temp with the "valve"...the thermostat, then the cooling system is plugged somewhere. Usually the tubes in the radiator!
If under hood temperatures are elevated to a state to cause other components to malfunction, I would definitely classify this as a coolant, or cooling system problem!
Stuart