Engine Overheating due to Radiator Fan Motor

In the link below it has two suggestion I see. The radiator fan has two speeds one was not working. They were not sure what was changed. They were told the fan has an idle speed and running speed. Could the fan speed control dropping out take out the temp gauge I do not know? The other idea, your car has a bleed plug to bleed the air out of the engine coolant system, air in the engine will make it overheat.
I did have a car one time and the fan relay would drop out after driving a while but to say it dropped out at a certain time, mine dropped out in slow moving traffic I think due to temperature. After engine cooled it would work for a while, I found the problem because when the fan quit working I wanted to check the fuse but touched the relay and it was very hot.
https://www.cargurus.com/Cars/Discussion-t12549_ds628075
The link below has some good info. It also has a comment that the temp sensor sends signal to PCM and PCM sends signal to control fan and signal to gauge.
https://www.justanswer.com/car/150qo-2001-pt-cruiser-cooling-fan-will-not-on.html
Well maybe this may help
https://www.youtube.com/watch?v=9U9eDsmRbqY
https://www.youtube.com/watch?v=url0zrCeyF4
One thing in the video was you have a low speed relay and high speed rely Just some thoughts or things to look at.

check idle speed control motor .idle air control

Found this to be common. Has a two speed fan. Fan also goes on and off at different speeds. Operates differently with ac on are off. Should not go past the half way mark while idling on hot days and should always drop to normal position while driving. Fluctuation should not be very much Air in system, bleeder on side of fill, not on overflow. Check fan high and low speed.

Check your public library they may even have access to a paid data base. Maybe this will help. With electric fans it's almost always the motor, It's a two speed fan that has three wires. The black is ground, the dark green is the hot for the low speed and the yellow is the hot for the high speed. Run jumpers from the battery to the fan connector. If the fan doesn't run the motor is bad.