The engine, at idle, should not be overheating with just the mechanical fan BY ITSELF as long as no a/c is running.
Sounds like a weak/failing mechanical fan clutch.
Try raising the RPMs when it starts to heat up at idle, the temp drops it is the fan clutch.
FSM fan clutch test:
"The cooling system must be in good condition prior to performing the test outlined below to ensure against excessively high coolant temperature.
CAUTION: Ensure that there is adequate fan blade clearance before drilling.
(1) Drill a 3.18 mm (1/8 in) diameter hole in the top center of the fan shroud.
(2) Insert a dial thermometer -18 degree to 105 degree C (0 degree to 220 degree F) with an 8 inch stem, or equivalent, through the hole in the shroud. Ensure that there is adequate clearance from the fan blades.
(3) Connect a tachometer and an engine ignition timing light (to be used as a strobe light).
(4) Block the air flow through the radiator by securing a sheet of plastic in front of the radiator (or air conditioning condenser). Use tape a the top to secure the plastic and ensure that the air flow is blocked.
(5) Ensure that the air conditioner, if equipped, is turned off.
WARNING: USE EXTREME CAUTION WHEN THE ENGINE IS OPERATING. DO NOT STAND IN DIRECT LINE WITH THE FAN. DO NOT PUT YOUR HANDS NEAR THE PULLEYS, BELTS, OR FAN. DO NOT WEAR LOOSE CLOTHING.
(6) Start the engine and operate at 2400 rpm with the timing light (strobe light) aimed at the fan blades. Within ten minutes the air temperature (indicated on the dial thermometer) should be 88 degrees C (190 degrees F). Satisfactory operation of the fan drive requires that it engage before or at 88 degree C (190 degree F). Engagement is distinguishable by a definite increase in the flow noise. The timing light will also indicate an increase in the speed of the fan.
(7) When the air temperature reaches 88 degrees C (190 degrees F) remove the plastic sheet. Satisfactory operation of the viscous fan requires the air temperature to drop 20 degrees F (11 degrees C) or more. A definite decrease of audible fan air flow noise should be noticed. Replace defective fan assemblies."