Bambi was not intended to be cute entertainment. Rather, in the natural order of things, deer are a prey animal. We are the wildcard evolution had not planned on. We manage our sustenance resources in a manner no other creature has. Unfortunately the overpopulation of species like deer happen because of this.
Predators are the natural population control system in nature.
The cruel thing we do to the deer is to let them die of starvation and disease. Not many of us like to think of shooting peaceful creatures such as deer. But these same people line up at steakhouses and order prime cuts of dead cows. I see no difference except in the fact that deer seem to have a better life in the wild up to the point of their death, whatever the cause.