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.<

The ECT on the early models that had the 105/115 On/Off
switch worked that way, but the units with a Thermistor [negative/temp ] do not complete the relay coil to ground .. They feed the ACC -N22 Control and that trips ground to the coil side [primary] of the relay via a transistor in the ACC control at the proper ohms .
So, the N22 is both the reader of the thermistor and the grounding switch for the coil of the relay...
Some S class cars have different resistor trip points [ usually V8s]
than do some of the smaller 124 cars. [ lower temp cut -in]
What the Mod does is parallel up the ECT to fool the N22 reader into thinking it is at trip time , as it only sees resistance [ mod actually puts the ECT out of spec.] , allowing a lower temp/resistance option...