Red Meat cancer is linked to the steroids and injections cattle are given to make them grow quick. this has been also linked to chickens that are farm raised by big processing firms. once again its a link. genetics and family history play a lot.
as far as deer , hog elk etc. my internal medicine doc and cardiologist eat the stuff i kill and like you i eat more deer and other game than beef .
fill your freezer with wild game and dont worry about it