Your question is temperature dependent. You can make the deboned tough by cooling it too fast. For instance, deboning an elk shortly after it dies and laying the chunks of meat in the snow bank. You will actually see the muscle fibers in the meat twitching and contracting from the cold. If left attached to the carcass as it cools, some of this contraction still tries to occur, but is hindered by attachment to the bones. The meat stays more tender. This is an extreme example.
By the time most people get to the animal, take their photos, etc., and begin to debone, much of this contraction has already occurred. No harm to the tenderness of the meat. Cool the meat in a reasonable amount of time, but don't hyper cool the meat like in the example above at least for a couple hours. For most people the deboning occurs in much warmer temperatures, and this won't cause any problems.
Thanks everyone for your input. I don't think I'll run into any problems as I didn't draw a single hunt I put in for! OTC EVERYTHING for me this year! GGrrr! Seriously, sounds like a toss up on the different meat care options.