Good points... let me work myself through this...  Let's consider my hunt last week.  Let's say I killed one right when I was going to leave on the night in question.   Now, I have an elk on the ground, it is still dropping heavy, wet snow, and I am already slightly hypothermic.  How would I handle that?
First, I would realize that I am in for a long night.  I would probably do some jumping jacks to warm up a bit.  I would build a simple shelter and start a fire.  I would drink some hot liquid to warm my core.   Then, I would get to work on the elk.  I would use the warmth of the elk to keep my hands warm.  The work of breaking it down would probably keep me warm.
If I was warm enough, I would break it all down and hang it from an improvised meat pole.
If I was too cold, I would gut it and break open the hips to cool the meat down.
In either case, I would probably leave the meat in the woods.  I would hike back to camp and sleep.  I would come back in the morning with an empty pack to starting hauling the meat out.
Once the meat is hanging, I personally would not haul a lot of meat in the dark especially if bushwacking.  It is going to take me several days to move the meat alone.  I don't see a real difference between it hanging at the kill site or hanging at camp.