When pressure in cylinder is very low (like when engine is idling), ignition timing, mixture etc. things have very little to do how the engine heats up - you can adjust both pretty alot to any direction and as long as engine keeps running relatively smoothly it does not heat up anymore when engine is idling - but only if cooling system is working properly.

But because there is no cooling air flow, problems with cooling system have very much to do how engine heats up while idling.
+in order to make fan work properly, there must be proper cooling liquid flow in the engine. Pressure loss, air in cooling system etc. will make fan much less effective and that is specially visible when radiator does not have lots of airflow and water pump speed is low.
For example, if there is air blockage in cooling system near fan switch - engine will boil before fan even starts working because switch is getting wrong info. And when you test fan, switch etc. everything seems to be fine.

When driving faster engine / water pump rpm is higher, both coolant and air flow through radiator is much bigger - problems with pressurizing, local boiling etc. are usually much less visible... but only before you have white smoke coming out from tailpipes

When pressure cylinder is high, mixture has very much to do what kind of temperatures are happening and where exactly. When cylinder pressure is high, too lean mixture will cause engine to heat up _locally_ (cylinder head, valves, pistons etc) which is usually not possible to see in temp gauge, at least before it is too late. Too much ignition advance, too much boost and some other things cause pretty much the same.
When mixture is too lean - temp raises - uncontrolled mixture burning called knocking starts - ruined piston and usually lots of other broken things too.
Very sometimes you will get early warning if engine temp is rising when pushing the engine at high load rate for a long time.

Many engines (specially charged) use extra rich mixture up top to cool engine down. Some engines use toluene, water etc. and separate injectors to do the same etc.

Of course, this is all very much simplified and guessing - like gunsmith telling you why your pistol is not running by listening your shooting
Hope this will still clear at least something.

Those were the times, when I was drilling small holes to thermostats... Do not do this at home unless you know 110% what are you doin; thermostat also has a big effect on cooling system pressure @ high rpm