The biggest danger from properly processed and cleaned wild game is parasites and some diseases not common in domestic animals.
Some of them you can get and never realize it, they just make you less healthy, while others are more noticeable.
Parasites is not a poll option.
I also would not want to eat deer with chronic wasting syndrome, aka scrapie relative, aka mad cow disease relative. No amount of cooking heat kills the prion responsible.
If you caught it the symptoms would most likely just be diagnosed as alzheimer's unless you were young.
If you get certain parasites they do permanent damage, others just damage intestines slightly creating scar tissue or reduce vitamin and other nutritional absorption.
The USA and its wild game is better in the zoonosis and parasite department than many parts of the world.
But there is still some things to watch out for.
I myself have never been sick, but if I eat game meat it is well done, and well done meat just isn't as good.