In an era where vehicles shut themselves off at stoplights to save six grams of fuel a day and LEDs are used to lighten the load on the alternator, it seems silly that having the radiator fans operate as soon as the ignition is turned on is correct. I would guess that indicate a problem in the PCM or its wiring. In fact, it used to be that if the fans turned on immediately, it indicated a fault mode of the PCM.
It could be beneficial to connect a scanner and monitor PCM inputs. This could reveal what the PCM thinks it is seeing, such as a coolant temperature in an incorrect range, or an AC pressure switch stuck closed, or a high transmission temperature.
But that may be a secondary problem. To maintain perspective, lots of vehicles with mechanical fans have been able to make and maintain coolant temperature with the fan moving constantly while relying solely upon the thermostat remaining fully closed until the coolant temperature reaches the proper level - Millions upon millions of vehicles, and for many decades.