The fan can be fused full time power but the relay that is controlled by temp and AC should be key on power, or fan would continue running after you parked the car/truck until heat was below temp threshold
Some vehicles had that setup, but usually had a timer with it so it could only run a few minutes after key off
Or there would be a dead battery in their future, lol
And never understand that thinking, when engine/key is off, there is NO coolant circulation, so cooling the rad coolant with the fan doesn't do much for the engine, lol
I guess maybe the fan in the engine bay would circulate some air, but it would be hot air after coming thru radiator, no, I never did see the benefit of that