You can test coolant fan by directly splice it to battery. Like you said, it is not normal to fault. Usually, there is a part called "fan activate switch" tends to be fault, which senses the temnperature. You can drive the car until warm and open the hood and let the engine idle. If temperature reaches high mark and fan not on, then the switch may be bad.
Turning key to ON or OFF is not right way to test it.
when very close to the red line. Acturally, it's pretty hard to reach red line unless rev the engine up to 6000RPM for a while or coolant system has problem.