If your setup is like mine, the fan should come on at 110 degrees, or anytime the A/C is operating.
The resistor in your pics just controls the speed (hi/low) at which the fan operates. I imagine it could fail in such a way that the fan would get no power to it. Mine (with green sensor) is supposed to operate at low speed at 105 degrees engine temp, and high speed at 115 degrees.
You can bypass the 105/115 degree sensor (or whatever yours is) by jumping the terminals at the sensor, which is located towards the rear of the head (on the intake side), and it should have a green or blue plastic top.
Jump sensor- see if fan starts.
If not, bypass resistor, and see if fan starts.
This rules in the resistor.
I speak like some sort of expert, when in reality I have not yet figured out why my own fan no longer operates at 105 degrees anymore (or even 115 for that matter), though it does operate when I jump the sensor.
Just have not gotten around to figuring it out...
Best of luck.
1986 300E 5-Speed 240k mi.