Here's what I do. Some people won't agree with it...but it's how I do it.
When I down a deer I find it. If it needs dragged to an area where I can work then I'll drag it. If it was a short quick trail I usually don't have to drag it anywhere.
I go to the truck, drive as close as I can. Grab the mid-sized cooler out of my truck that has about a gallon of cool water in it. Take the cooler to the deer. Bone the deer out right there on the ground. No gutting. Do the front left quarter, grab some of the neck meat on that side. Do the left backstrap, do the left rear quarter. Flip and repeat on the other side. Getting the fillets has a slight learning curve to it but it can be done. Get in there and you'll feel em. The rest is pretty straightforward.
All the meat goes in the cooler. The cooler is then carried (okay, honestly...it gets dragged alot, but it makes a nice seat while you rest) to the truck. On the way home I get a bag or two of ice. When I get home I fill the whole thing up with water. You can let it sit in that cooler as long as you want SO LONG AS THERE IS STILL SOME ICE IN THE WATER! Change out the water several times and add new ice as needed. I've left meat this way for up to a week. That water is colder than your frig. I've cooked backstraps literally 10 or 15 minutes after the animal died. I can't tell a difference but hanging the meat and letting sit in water, that you keep fresh, would accomplish the same thing...so do that as long as you'd like.
I'm on my own property so I don't have to worry about the carcass. (Bones and gust together, wrapped in a tangle of hide.) We've got more than enough coyotes to take it. It's been years since I fooled with anuses and esophagus. Every now and then I "dissect" a little out of curiosity to look at the heart/lungs/trachea.
Some feel that this method wastes meat. I'll admit you probably loose about a pound off the rear quarters in an effort to stay away from the butt hole. One buddy gave me quite a hard time about it. The following week I watched/helped him clean a "hung" and skinned deer. When he acted like we were done I quickly got another two pounds of meat off the carcass and told him to never "lecture" me on being wasteful.
(Granted, it was sausage or jerky meat.)
If I ever learn or want to learn to tan the hide I'll have to go back to the old way.