It all depends on the weather.
Deer hide and hair is chock full of bacteria and stuff.
I leave it hang with the hide on if it is going to stay cool and dry.
I take the hide off if it will be warm, or rainy. When the hide comes off, I butcher the deer.
The hide is much easier to remove while the deer is still warm. I would not let the deer hang hideless because the meat will dry out.
Aging the meat is a whole other matter, and can be done in the cooler or refrigerator, before or after freezing the meat. So, I can kill a deer some days before going home, and age it on the rack or in the cooler. Then freeze.
Or, I could kill a deer on the last day, butcher it, freeze it at home, and age it when I pull it out of the freezer and place it in the fridge. On the whole, conditions are most controlled when I age the meat in my fridge.
That is, if I don't miss.
Killdeer