Would it not be simpler to buy ice bags in town, and take them out to your home, or camp? Then, build a simple tent, using a garbage bag, or visqueen ( or an old shower curtain) to keep the cold air close and around the deer carcass, tying the bag if ice above the carcass, and suspending anther inside the chest cavity? If you have air conditioning- I know that is hard in camp, but often the shaded areas during the Fall are substantially cooler, and stay cooler-- you should be able to keep the meat in good condition while you process the quarters, one at a time, and move the meat into ice chests and coolers. I watched a couple of guys set up a table inside an outbuilding at a farm where I was hunting, and they processed both their deer while the rest of us were out hunting. The only thing they did that I would not do was fail to wash the meat off, and dry it before wrapping it. But, as they finished the packages, the packages went into an ice chest, and that is how the meat was transported home. Bones and the rest were left in the garbage pile at the farm. There was water available either in the building, or just outside- I don't remember the inside sink now that I am trying.
Bacteria do exist in the air, and elsewhere. But, sawing bones just makes for very tiny niches where they can stay, and breed and multiply, feeding on the rich bone marrow that is in the bone. Its the multiplying that creases the spoilage in meat. Bacteria need food, water and temperature. By cleaning the blood off and drying the meat before storing, you deprive the bacteria of a quick, and convenient source of water. The missing blood also would have provided a rich source of food for bacteria.
There is no point to arguing something that can be easily proven in a lab, or even doing a simple test at your own home. Take two similar size and shapes of meat, one that has been washed of blood and dried, and place it wherever you desire next to the second piece that has not been washed and dried. See which begins to spoil first.