This obviously (to me at least) another of those things where many people are looking for THE RIGHT ANSWER when there are many different circumstances, each requiring different answers; and where the "right" answers in one situation are often quit disatrously WRONG in others.
One factor, offered for consideration, is that many experienced "animal persons" assert that kittens who are not brought live prey by the mother almost never learn to be effective hunters. It's difficult in many circumstances to know whether a kitten was "taught by mommy" when pets are obtained from unknown sources; but some breeders, and many "housepet owners" do have a tendency to sequester mother and kittens at least for and immediately after birth. My own observation is that there is some truth to the assertion that kittens so raised will seldom be effective hunters but may be somewhat more inclined to catch prey, perhaps by accident, to "play with" and less likely to consume what they catch.
Urban pets are likely to be included in the "interested but incompetent" category as hunters (good prospects for management jobs?). They probably should not compete much with wild predators, as the few wild ones likely to be found in urban areas are of kinds that have learned to associate with humans and are reasonably successful in that environment.
Barnyard kittens may be expected to have gotten the lessons from the mother cat, and are quite likely to be efficient and effective hunters. They also are more likely to have access to areas where wild prey is more accessible. If numbers are kept to reasonable levels, the kinds of prey they're most likely to seek and catch are mostly the "varmints" that farmers need help with controlling.
Feral, or "semi-wild" country cats probably are the ones most likely to have a significant impact on wild populations, and particularly on "game" populations. Predation on game birds, many of which do nest on the ground, can be quite severe; although most of the destruction comes from destruction of eggs and/or hatchlings still in the nest.
Even large-adult-feral-hunter-cats are unlikely to be able to kill many adult rabbits, and fewer still can successfully catch and kill an adult squirrel; but they may be able to drive the adult away from a nest and kill all the babies in either case.
The recommendation of game wardens in my area has been, for several decades, that any cat "50 yards from the farmhouse1" should be shot if it can be done safely, since they are destructive of game bird populations due to predation on nesting birds (and the most popular of the game birds in the area are ground nesters). For comparison, the same game authorities will tell you to "ask the farmer" about other kinds of "varmints." Some farmers like their skunk populations and others seem to prefer that one leaves the 'possums alone. Most farmers I've met would prefer it if you could shoot the crows, but since they're "migratory birds" that's illegal.
1 It's generally illegal to discharge a hunting gun within 50 yards of an occupiable structure, so they're saying if you can, shoot.
These anectdotes contribute little to explaining any particular situation, but my point is simply that there is no "one-size-fits-all" problem, much less a single "solution."