Here in NM the ML season is or was in late Sep. Daytime temperatures were in the mid to high 70s where I hunted. If I killed a deer on Sat I opened it, hung it, and made sure I got it to town on Monday. Never did I have a hint of spoilage.
At an Army school I asked a vet about meat spoilage, and he said if you get the body heat out by cleaning and propping open you can eat the meat safely as long as you can stand the scent.
I always tried to hang my deer for 5 days, and it was never at "refrig" temperatures. The hanging made the meat better.
So I conclude that the requirements for restaurents are good, but not necessary for hunting conditions.
Clean the deer immediately, prop open the carcass, try to keep it in the shade, and don't panic but get it to the shop when you can within a couple or three days.
I never let the meat contact water. The old saw that I you do penetrate the guts or stomach you contaminate the meat is baloney. There is no blood flowing to spread any contamination. Just clean it off and take care of the meat.
You can use a wet cloth to clean it off.
Elk are a different matter as they are so large they do not cool as rapidly as deer. Try to get it off the ground and quarter it and place in bags.
We ate every deer I killed, and that was a lot over a 50 year period, and all were good with never a hint of spoilage.
Ecclesiastes 12:13 *¶Let us hear the conclusion of the whole matter: Fear God, and keep his commandments: for this is the whole duty of man.
14 *For God shall bring every work into judgment, with every secret thing, whether it be good, or whether it be evil.