The fan cools the contents of the radiator.
The engine water pump pulls those contents into the engine and returns heated coolant to the radiator.
The electric fan will only run when it senses hot coolant and the electric fan has electric-power (the key must be on.). But if the engine is not running the cooled coolant will not cool the engine. Instead, the fan will only cool the radiator coolant, and the non-running engine will still contain hot coolant.
Bottom line: If your engine is not running then the fan might run short or long until the temp sensor detects sufficiently-lowered temperature, which will NOT benefit your engine unless the engine is running.
When you ran your RTV “for a couple miles” it didn’t overheat because 1- the coolant was being cooled in the radiator by relative wind due to your forward motion and 2-the engine water pump was pulling cooled coolant thru your engine. So no useful info is obtained by that activity as regards your electric fan. If your engine is not running and the fan runs only a second or two it could be any one of several reasons -1- the coolant was on the threshold of the sensor, 2- the sensor is defective, 3- the connection-terminal at the sensor or elsewhere is intermittent, 4- the fan motor electric supply is inconsistent due to loose/corroded connections, etc. and on and on and on.
If your key is on, and your fan is running, it does not benefit your engine or your battery because coolant is not being circulated thru your hot engine and your battery is not being recharged while it is discharging thru your fan motor.
When your engine/coolant is hot you should run your engine at speed (not idle) and if possible your RTV should be moving into the wind. Otherwise, shut it down and turn off the key and let it cool down.