The temp switch on the filter/dryer requires that the refrigerant pressure be high enough to activate it. If you're low on refrigerant or in a low ambient condition, it's normal that the fans won't be running. Since they run when you short the wires together, that proves the fans, the resistor, and the relay that controls them functions as designed.

The electric fans are not there to cool your engine, they're there to improve performance of the A/C and to serve as a backup if the engine is dramatically overheating.

The grey temp switch doesn't enable the high fan speed until something like 115˚C.

Make sure the condenser coil and the radiator are clean and not full of dirt or debris. If you haven't done it yet, or have no record of it being done, it's time for a fan clutch. The engine driven fan is your primary cooling air flow. A weak or failed clutch will dramatically reduce the cooling ability, especially in city or stop/go traffic.

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Current stable:
1995 E320 149K (Nancy)
1983 500SL 120K (SLoL)

Black Sheep:
1985 524TD 167K (TotalDumpster™)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)