High Limit Switch

Some of the older Goodman furnaces was equipped with a flame sensor it was just a solid metal probe with one attached wire and when it heated up it caused a small millivolts in the probe if this sensor fails your furnace will not light or will go out after a few seconds, only maintenance required is cleaning the metal probe with steel wool or sand paper and check for a loose terminal due to heat stress its cheap and inexpensive to replace so it should be replaced every 5 years most problems with these probes were burned wires and connections as they are directly in contact with the burner (NOTE) Goodman furnaces were designed to run a few seconds in order to give the flame sensor time to heat up and produce the electrical current to activate the switch if no activity is sensed the burner will shut off, a flame sensor works on the same principal as an expansion thermo coupling except there is no movement on a flame sensor it works on millivolts of current generated by heating metal where as a thermo coupling has a mechanical rod in the housing that moves with heat expansion and stays as long as the pilot is burning,

I believe you are correct that the fan itself is the problem.

Your diagnostic test of supplying power to just the fan, and the fact that the symptoms stayed with the fan, sound like conclusive evidence.