I may be wierd but I have never minded after the shot chores

. Well unless gut shot...then YUK...[:'(].
I use a 3.5" drop blade and zipper the hideup to the rib cage, then split the pelvis and release theesophagus, diaphram, connective tissuesand dump them wherethey lay by pulling from the esophagus to anus (leaves a nice clean track inside, less messand full gut pile if done properly). Finally remove the tarsal/scent glands and disgard (this knife is now out of commission until it can be washed). I now tag the animal and the rest ofthe job is completed at home by a light rinse with clean cool water for the inner cavity and skin immediately.
I usually let it hang for 24 hours then debone for the freezer. If to warm of temp I quarter and fridge age it for a couple days before cut and wrapping.
My tip is get the carcass cooling asap, this means field dressing, etc. If you have the luxury of a walk in cooler or meat locker the skin can be either removed asap or left on. If not and the temp is above 40, i'd skin asap and let the entire carcass get cooled then deal with the meat how you choose.