First thing we do after tagging the animal in cold weather is to build a big fire. That helps quite a bit but it is still miserable. I think boning out a critter might be the better way although I've never tried it in cold weather. Saw a guy on a video break down a cow elk that way. Took him ten minutes.I muzzleloaded a doe at -25 here in MO about 30 years ago. Never again.
She was almost getting frozen after gutting and skinning. Cut her in to pieces and took her in to the house.
When I was younger I HAD to hunt. Now I am smart enough to know that is dangerous weather.