My uncle ran a butcher shop on his farm for 60 years. When I killed my first deer, age twelve, I asked him how long I should let it hang. My uncle walked me to his cooler and asked how many animals I could see with their hides still on, I replied none. My uncle stated that you take the hide off an animal as fast as you can and that includes deer, bear, and moose. The glands are in the hide and will secrete the oils back into the meat if the animal is allowed to hang with the hide on.
Since that revelation I have always removed the hide ASAP, people who dislike venison will comment on my venison tasting better. I will age deer, hide off, and quartered in the extra (beer) fridge for a couple of days but do not hang it with the hide on.
I usually eat the heart the same night and on other ocassions have had tenderloin the same day. I have no issue with eating the animal while there is temperature left in it, i do believe I am too busy to cook meet at the kill site.