My dogs sometimes kill wild rabbits, raccoons, possums, and try to run down deer. Most wouldn't have a problem with that anymore than they do cats killing mice, rats, voles, gophers, and sparrows. Dogs just tend to kill things we are trying to raise like chickens or harass larger livestock in to injury. I've only had 1 stray/feral cat go after bantams and caged quail and it's dead. The rest I ignore unless they take up residence and start reproducing. Then they get neutered or pts. If dogs only went after wildlife we would probably just consider them an annoyance and some wouldn't even care. If they did no harm to my livestock or other pets my goal would be to capture them and turn them in to the humane society or somewhere they can find a more responsible home like I do with cats I don't want around. Unfortunately most dogs are a danger to my poultry, rabbits, cats, other dogs or even people so they get a stronger response. If you want to start shooting roaming cats for killing things in your yard only a few are likely to complain and it will probably be the same few that complain about shooting dogs. I just don't think too many see it as a priority to protect wildlife.
Most of the cats I have are just because they'd be dead otherwise. I've tried to rehome one numerous times. I can't keep them as indoor only and I haven't decided any wildlife they've caught come above their lives. Plus one can't find her way out of a paper bag so her impact on wildlife is limited to the mice who run right in to her. I found her on the center line of the highway as a 3week old kitten with a rock indent in her head and decided at minimum she would die a less painful death than laying there on the hot blacktop bleeding to death. Surprisingly she lived so I gained another indoor/outdoor cat. Dogs are a choice, cats just happen. I'm not responsible for their existence on this property.