Geez. What I suggested is simply a way to test if the fan works, that takes a few seconds, as the OP was asking.
If the electric fan doesn't come on when you shorten out the 2 connectors with a screwdriver etc, the temperature sensor is the likely culprit (see post #18)
Shortening out the A/C pressure switch should do the same, but I would check the temperature sensor first.
If neither of the 2 methods work, there is also a fusible link for the electric fans power supply if I remember correctly.....
If you have a remote infra red thermometer, its fairly easy to check the radiator for cold spots. If a new one is needed, I think Denmark's Nissens makes the radiator with the reinforced neck. On a BEHR, the little plastic pipe on the side of the neck breaks quite easy (don't ask how I know).
A new thermostat is cheap, and if the old tests ok in boiling water, you'll have a spare.
I don't think anyone here has mentioned to replace the overflow containers pressure cap.
Its an even cheaper item, an old/weak spring can reduce the function = reduce system pressure with corresponding increase in coolant temperature.
BTW, If you ever go through your whole cooling system with new water pump, hoses (with quality clamps), thermostat etc, just before installing is the perfect time for a de-oiling flush (can be done anytime), followed by a citric acid flush to remove calcification/scale. If you don't know the age of your water pump, forget the citric acid, it might finish it off.
Edit: Just read John350 post #43. Disregard the A/C pressure switch procedure.