This sounds like a problem with the fan clutch. If the fan wasn't working, it would overheat all the time (you indicated the fan cools it down). If the fan is turning too fast when it's cold, it may prevent the engine from properly warming up.
If you look at the pics below, they provide directions and a picture for testing the fan clutch. Do me a favor and perform the test.
Let me know what you find or if you have other questions.
See pics below
Images (Click to enlarge)
Friday, March 26th, 2021 AT 7:36 PM