One can use "harvesting" and it is correct, the same as if one had used "killing." However, neither are an accurate representation of the act of hunting.
I wonder how far back "harvest" has been used to refer to the act of hunting.
I bet it transcends any of your guys' attempts to turn it into some sort of liberal academia-originated conspiracy.
Harvest is a very artful word and implies a skilled investment of labor to obtain a tangible good, brought forth from nature for man's consumption. In that light, how is a successful hunt, not a harvest?
It's association with Death is quite natural too. Harvests come in autumn as everything winds down and winter cold descends. We pluck sustenance from plants that grow from the earth and sever its ability to live, so that we might draw life from it through the winter. The Reaper even carries a scythe with which he harvests the souls of those he has been sent to claim.
It wouldn't shock me in the least to read some 17th or 18th century text and see a reference to a successful hunt as a harvest.