I think a lot of these folks underestimate what it takes to hunt an elk, field dress it, bring it back to their vehicle, and put away 300 lbs of meat. Even if you borrow a gun as one person interviewed here would, you need a truck, knives, a bone saw, camouflage, maybe a tree stand, and a large freezer, among other things. It is reasonable to hunt for food, but it's not reasonable for your typical "hospital manager"…
The maximum effective range of a bow is 40 yards (above that, and the deer jumps away before the arrow gets there). Shotguns are a little bit higher, but even so, I have never heard of a bowhunter shooting another bowhunter in the area -- one time a novice bowhunter illegally hunting in a park shot a dog, but that's about it.
When you shoot a deer, you do not chase it. You wait for it to expire and then look for it. This is taught in the bowhunting safety course.
The real difficulty is that even in areas where they are overpopulated, there are at most one group of deer per 60 acres, and a bowhunter can barely cover 1/4th acre. Deer do not like to move around, don't follow the same routes every day, and are quite cautious and sensitive to odors and noise. They also spend most of their day in areas such as thickets where a bowhunter cannot get close to them without scaring them off.
So unless the deer happens to walk by the hunter, not smell him, not get startled by his movements drawing a bow, and is facing the proper direction to offer a shot, the hunter will go home empty handed.