It’s probably designed so that if the temp sensor is accidentally unplugged, the fan will spin. That’s the way it works on my early 90’s Corolla. That’s a fail-safe mechanism. It’s better to have a too-cold engine than a too-hot one.
On my car, the fan spins when the engine main relay is “on”, and the fan relay is “on”. The fan relay is enabled (again, on my car) by the radiator fan temperature switch. The ECM isn’t involved. If that switch becomes enabled by the coolant temp, it will turn on the fan relay, and the fan will spin.
On newer cars, probably the case for a 2001 like yours, it works differently. The ECM reads the coolant temp and decides for itself to turn the fan “on” or “off”. It seems like the fan should start if the sensor remains unplugged and the car is started and the engine is running. I expect that’s a symptom of something being wrong.
I guess what I’d do in this situation is (1) bench test the coolant temp sensor that the ECM uses for this function. It should read a certain number of ohms, which change according to the temp. You’d put it in a pan of water and slowly heat it up and read the ohms vs temp. If it is working it will correspond to the specs for the part. (Which would be in the shop manual.)
If that tested out ok, I’d test the ECM signal that turns on the fan relay. It should turn on if the temp sensor is disconnected. If it is in fact turning on, but the fan isn’t turning on, that should prove easy to fix. It’s either a bad fan relay or a connection gone bad somewhere in between.
If the ECM isn’t turning the signal on, then either the wiring between the sensor is bad, or there’s a problem with the ECM. Either there’s a bad connection nor it needs to be replaced.
Another thing you can do is turn the AC on. That usually causes the radiator fan to spin too. Does the ECM send out the signal for the fan to spin when the AC is turned on?