"That beloved pets should not be given the opportunity to kill wildlife for sport is the easy part of this debate." Why not start by extending that to humans as well?
In many cases, humans are hunting species which don't have any effective natural predators, either because said species were introduced into the environment by humans (as is the case with feral pigs), or because humans have exterminated the existing predators (as is the case with wolves, who kept deer populations in check).
For example, in Texas, the problem with feral pigs is so severe that two thirds of the entire population needs to be culled each year in order to maintain it at a stable level.
The things that cats hunt, though, generally do have other natural predators to keep them in check, and getting cats into that game causes populations to plummet.