It was Mother Nature that made us omnivores, so why would She be "banging her head against a tree"?
So, yeah, boars hunt and eat deer, while humans hunt and eat them both (Top of the food chain, and all). Many cultures, and the Amazons would be one of them, survived by hunting. A difference between the Pagan and Monotheistic religions is that the Pagan religions saw everything as connected, and sought to be in tune with that, with a respect and veneration of Nature. The great Monotheistic religions viewed Man and Nature as separate, and Nature something to be overcome.
If Diana had done as instructed, the deer's death would have been quick and far less painful. The deer was going to die, regardless.