[93celicaconv]

The switch that turns your fan on if your coolant temperature is getting high is a coolant temperature switch. I don't have information on where that switch is on a Corolla 4A-F engine, but on a Celica 4A-FE engine, that switch is at the bottom of the radiator. If you disconnect the wiring harness from that coolant temperature switch, the fan should turn on at high speed. If your coolant hasn't reached that elevated temperature, the switch will not open, causing the fan to turn on. In normal driving, the fan will not turn on as the car is moving and has air going through the radiator cooling the coolant. If the car is parked with the engine running, there is no air going through the radiator to cool the coolant, so it is normal after idling while parked for 10-20 minutes for the fan to start cycling on & off. If yours does this, you don't have a problem to solve.

 

[93celicaconv]

My fan never turns on unless the harness is unplugged from the fan switch (even on a 40*C day in heavy traffic),

Does your coolant temperature gauge ever start climbing in a situation like this? You might want to leave your engine idling while parked and watch both your fan and your coolant temperature gauge. If the coolant temperature gauge begins to climb above it's normal operating temperature without the fan kicking on, then you will have confirmed the fan issue. If the fan kicks on after 15-20 minutes without the temperature gauge moving above its normal operating temperature, then the fan control is working normally.

It looks to me that your wiring harness connector has a protective cover over it (if that wiring harness connector is not used on your particular car). But that said, if you move it and the fan turns on and off rapidly, I would agree that moving the wiring harness may be causing it and that you have some kind of wiring issue there. What are the colors of the wires on that connector? Include the color of the stripes on the casings of the wires if they have any stripes on them.

 

[93celicaconv]

Great. Does the plug (cap) have metal terminals on it? If yes, it may be a jumper plug (meaning when it is plugged in, you are jumpering the wiring terminals which keeps the fans in an OFF state. If you pull the plug out, you are creating an open circuit in that wiring harness which turns the fans on. If this is the case, you should clean the terminals on the plug and spray an electric contract cleaner on the connectors on the wiring harness connector side and (If possible) squeeze those connectors in the harness to close the gaps a little (puts more pressure on the cap terminals). That may help resolve the problem you might have with corrosion in there.