Slight correction in your basic oxford English dictionary buddy:
engine warms up = OVERHEAT
certain level = IDEAL TEMPERATURE
I am driving in the desert for over 10 years and almost all the cars when pushed for dune bashing (not desert safari - train track), do get slight overheat and this fan clutch produces a hell of a noise that felt inside the car cabin. I have heard this on my Land Cruiser, Patrol, Discovery, Range Rover, Cayenne, Pajero 3.0 and even on this same Pajero 3.5 for last 7 years.
The radiator has been detailed clean 4-5 months back, as soon as summer started and I have driven the same Pajero for all the summer drives. It raises the temperature and again brings it down in 3-5 minutes provided clutch fan is working as it is intended.
Time to refresh your car knowledge too:
Source: https://en.wikipedia.org/wiki/Fan_clutch
When the engine is cool or even at normal operating temperature, the fan clutch partially disengages the engine's mechanically driven radiator cooling fan, generally located at the front of the water pump and driven by a belt and pulley connected to the engine's crankshaft. This saves power, since the engine does not have to fully drive the fan.
However, if engine temperature rises above the clutch's engagement temperature setting, the fan becomes fully engaged, thus drawing a higher volume of ambient air through the vehicle's radiator, which in turn serves to maintain or lower the engine coolant temperature to an acceptable level.
Check the below video of comparing the new clutch failing when the car was overheating vs below video of old clutch modified as the direct clutch now to run all-time at engine RPM.
If the new clutch was working or really genuine, then it should have been running at below fan speed and not so slow to bring the temperature down.
Play the video at high volume to hear both channel audio (slow and high-speed gushing).