There are two prerequisite conditions that contribute to meat going bad because of yeast, fungus, mold, and bacteria growth--wet AND warm. There is one condition flesh to rot, and that is warm. Those are the two things that are potentially happening inside the cooler.
Bacteria and such can only grow if the meat wet AND warm. Also, where is the bacteria growing? It's only on the outside of our ungulates. That's why we can eat rare red meat, the outside is cooked to kill bacteria and the inside has been "sealed away". We can't eat "rare" chicken, fish, pig, bear, etc. because there is bacteria, worms, or other nasties on the inside of the flesh.
Meat rots if there is warmth allowing the cells to breakdown. That's "bone spoil" and straight up rotting that will make you puke. We all know the smell of rot. Rotting starts about 4 minutes after death. The blood stops pumping and cells start to build up carbon dioxide, which is acidic. Cell membranes start to burst and the flesh breaks down without bacteria involvement. I didn't find anything on deer with a quick google search, but this is all about human decomposition...
https://www.aftermath.com/content/human-decomposition/
Besides rot, there is also flies, mold, yeast, fungus, and bacteria that contribute to the decay.
To prevent nasties from growing on meat, all you have to do is make sure meat isn't both wet AND warm. Meat can be cold and wet, so sitting in ice water it will marinate in the water, drain color, and change texture, but that weird colored meat isn't spoiled. If the water isn't changed it can make the meat taste gamey and pull out flavor, so if you can, avoid that for steaks, but you can put it in burger.
Or, meat can be "warm" and dry, which is why we like to get it hung up and a crust on it ASAP in the field. We can't always get it cold, but if it is warm that can make it dry. That dry meat is also "edible" for a while, but because it has gotten crusty and usually has hair and other contaminants, we cut it off and throw it out.
If it is a warm wet climate where you have rain and it is hard to get it to dry, you can also inhibit bacteria growth by putting salt, pepper, citric acid, etc. on the outside of the meat to make the surface less hospitable.
To prevent rot, you just need to keep it below whatever that temperature that is, to stop or slow down the process. If you can get it cooled down from body temp fast enough, ambient temp in the shade for a day is enough, and hanging in a garage where it is cool is sufficient to stop the cells from breaking down because it arrests the cellular breakdown. I think that the dry aging process might allow some of that cellular breakdown to occur and to let the meat dry. I don't know that aging wild game meat for an extended time has the same benefit of aging beef cows with all the fat and moisture in them.
The problem with meat on top of ice or frozen jugs in a cooler is that the direct sunlight or hot temps is enough to penetrate the insulation and heat the air underneath the lid and above the ice enough for bacteria to grow and the cellular decay to continue in meat that isn't actually in the ice. I didn't realize this, I though the cooler kept everything cool inside.
Test it out, put your cooler in the sun, and then lift the lid and feel the inside/bottom of the lid. It can be very hot here in AZ, so I know what that is like. And, I let some meat get away from me like this, even though there was always ice in the cooler. I spent way too much time thinking about how it happened, lol.
On the other hand, I have kept meat directly in ice water and covered with ice for a couple weeks. And, have kept processed meat in ziploc baggies in ice for a few days as well. As long as all the meat was covered with ice or in the icy cold water, it was good. A couple times there was only a few pieces of ice left, but the water was still very cold. I had no spoilage.
Cool air sinks, so the ice can stay frozen and cold, but if the meat is not covered with ice on top, then wet meat is exposed in the much warmer air at the top of the cooler. Strangely, the meat acts as an insulator to keep the ice cold more than the ice acts to keep the meat cold...
A cooler is very inefficient, so it can be warm on top and icy cold at the bottom. Its why we reach for the cold beverage in the ice water and avoid the barely cold cans on top of the cooler.
When that happens you (I) have created a situation where parts of the meat can warm up enough for bacteria to grow and meat can begin to slowly rot.
If meat is rotten so much to spoil it and definitely make you sick, you should be able to smell it if you cut off the crusty nasty exterior. It will smell bad on the outside and inside. It can also have a different texture. Flesh basically rots from the inside out because that is where it is the warmest and the cycle of cell degeneration is accelerating. No amount of cooking will make it edible. Humans are specially evolved to have a sensitivity and repulsion to the smell of rotting flesh and very high disgust response for a reason...
If the meat has not rotted, then with a little bacteria/yeast/etc. growth, you should be ok with cutting it off and cooking the meat. Remember, the bacteria can only grow on the outside. I'm no expert on all the possible types of bacteria and such, so do your own research if you have cultured up a nasty rainbow of fuzz. But, a friend had a refrigerator go down, and he had some of the white growth going but no rotting. We trimmed it up, froze it and our families at it up.
When it doubt, throw it out. But, a little information can help you make a little better decision than just straight up fear.
. His theory on uneven cooling in coolers is correct. Meat covered in layers of ice are chilled more evenly in a cooler, and meat hung to dry and crust is dry aging.
