Our cars (1986 300E) do not have a "module".
The sensor drops its resistance as temperature rises. Once the resistance is "low enough" the coil of relay K9 is grounded (via the sensor), this "pulls" the contactor and closes the circuit, which sends 12 volts to the auxiliary fan via fuse D (18 Amps).
The fact that grounding the sensor connector makes the fan work tells me that the relay K9 is good. This relay would be the "module" that makes the 'decision' of turning the fan on.

"My new theory is that whatever component measures the resistance value must be faulty."

Unfortunately, that would be the sensor.


Remember, the sensor that reads the engine temperature for your instrument cluster (the last one on the head) is not the same that drops its resistance with temperature for the auxiliary fan (the middle one on the head?). (Although some models have the 4 pin type, which includes both)

I would follow the troubleshooting indicated by Arthur Dalton, and JimF.
It appears one or more of the wires is so brittle and corroded the circuit "thinks" (high resistance) is open unless grounded directly.

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1999 Porsche 996 Carrera Convertible
1994 420E - SOLD
1986 300E - SOLD, what a car
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