If my dog ran after livestock I would keep my dog away from it. I couldn't see myself killing my dog because of something like that.
Have you ever lived in farm country? The reason I ask is that chasing livestock is a crime in many rural areas. the owner of the dog will be liable for any damage done to livestock---including bruising damage and potentially loss of gains if the farmer can prove it. Not only that its perfectly legal for a farmer to shoot a dog chasing their livestock as a means of protecting the livestock. My grandfather shot my neighbors dog when I was a kid and took the dog to their front porch--he'd warned them multiple times about keeping the dog up, and when they didnt, he shot the dog. They werent happy, but legally, there was absolutely nothing they could do because the dog was chasing livestock on my families farm property.
Also, I'm not talking about just barking at some cows in a pasture--thats annoying--I'm talking about getting in with the livestock and running the cattle until the cattle (or sheep, or goats, or pigs) get out or get injured or worse yet, actually killing a calf or something like that. That is one of the things that a dog just doesn't do, unless its under direct control of the owner at the time its doing it.
I'd much rather be the one killing my dog---humanely---than the risk of some one missing the shot and gut shooting the dog or blowing off a leg and the dog laying in a ditch dying. Its typically a prey drive thrill of the chase moment thing that given a chance once they've done it, they will do it again and again.