I'm probably not the guy you want an answer from being that I haven't done this... Great idea, however.
I would imagine you should be able to get away with using just underlayment underneath 1/2'' drywall. Studs at 16'' O.C. should be close enough to not allow for much pull from a whitetail shoulder mount. You may want to still use one of those plastic/metal expanding inserts though if just using underlayment. I have hung deer heads on a paneling wall w/o a stud before using those inserts. Otherwise, either 1/2'' OSB or plywood should work perfectly fine for just a basic screw.
Keep in mind, your outlets will need to be nailed to the studs a little further out to compensate for the extra thickness.