I don't use cotton game bags for the same reason I don't wear cotton shirts in the backcountry. The big difference with what you mention is that meat isn't pumping out armpit sweat, so synthetic is a good option for me. I use them because I never know if I will get the animal to the truck in 6 hours or 4 days and cotton just doesn't breathe or dry quickly in addition to the extra bulk and weight. A cotton bag will stay wet a long time which is very bad unless you can quickly get the meat below 40 degrees.
If I need to go a couple of days, it is simple to rinse the blood out of the bag, air dry it for a few minutes and rebag the meat. Repeat this until the meat has formed a crust and stops leaking and you have a nearly foolproof system that keeps everything clean, dry and cool.
To each their own.