I have a PS2012 power supply where the fan is always running from the moment you turn on the power supply. Otherwise, it seems to work OK. I am assuming it has some thermal switch that has failed. Is this just a noisy nusance or is it harmful to the power supply for it to run all the time? I would not think so but then I am not an expert on power supplies.
Can never have too much cooling. It won’t hurt anything. If it’s loud though, might be worth having repaired just so it’s not annoying.
–Randy