Admirable idea but their numbers are astronomical and predation will continue as long as they are alive.
Kinda like saying don't shoot wild pigs, coyotes and deer, just tranquilize and neuter them and all will be ok.........
Wild pigs unfortunately are a threat to habitats and destroying more than just other animals ability to live, they are destroying farm land. I live in a state in the USA that has a major feral pig problem.
Deer hunting in every state is closely regulated with specific seasons, not indiscriminately killing of Deer and Deer are not Domesticated animals, never have been. Deer do carry diseases that (i.e. Lime Disease) that are dangerous to humans.