Change the cap so that issue is gone. Then look for other leaks.
The water pump can leak at the seal weep hole and yet still not overheat in use, you still change water pump to fix it.
Every car engine on the planet gets somewhat hotter when it shuts off, that is what is known to OEMs as a heat soak. The engine gets hotter for a bit as the water pump has quit turning. Any mechanical water temp gauge that still works with power off will show it, the engine say at 220 degrees water temp running then goes up for a few minutes to say 250, the pressure comes up too. Normal, the system has to put up with it with no leaks. The cooling has the coolant at slightly cooler than engine metal is, when it quits flowing then the engine metal evens the coolant temp up to exactly what it is. In normal use coolant say at 220 but engine metal at around 250 or higher, the oil temp, which is commonly higher than water temp. Engine temp will even be over that a bit, as the oil is a coolant of sorts too. They all work to even out as soon as engine stops, then engine metal acting as a heater for a bit until even then all cools down at the same time.
Fix the driver first, then fixing the car becomes so much easier..........