The use of either or was dicussed on the political forum about 5 years ago. It got pretty heated.

I see it as either one being correct depending on the person and their involvement in game management.
If your are a person who does nothing to help manage game andall you do is hunt you are IMO more correct to say you just hunt and kill game as a result of your hunting activity.
If you are involved in game management such as Quality Deer Management you harvest deer with a goal on mind for the entire herd. You are managing or influencing the raising of deer using specific practices to achiever a goal whether it be size or numbers or ratios.
You do not have to be in the "business", run a deer farm or fenced in operation.
The deer being wild has no influence on the use of the word. You don't kill wild berries,you pick them and you are therefore harvesting them.
Either way you still have to kill the deer to harvest it.