This.
My first dog as an adult was a blue tick coonhound mix I adopted from the Humane Society who was dumped there because he was fearful by nature and 'wouldn't hunt.' A decent human being doesn't just kill a dog that won't hunt - they give it a chance to be somebody's pet, because there are far more people who want a dog to snuggle with on the sofa and love than to keep out in a kennel in all kinds of weather and take bear or coon hunting on occasion.
A dog that kills livestock can technically be killed by law anywhere in these United States, I would reckon. Certainly they can be killed here in liberal Massachusetts, so I can't imagine it is illegal anywhere else to kill a dog that is worrying (mauling) one's livestock. When I was a kid my uncle who lived on my great-grandparents' farm shot and killed a few of the neighborhood dogs who had mauled and killed some of his sheep while running in a pack, as friendly pet dogs will do - instincts don't get washed out by sleeping on the sofa or eating out of a can or bag. The fault was entirely the dog owners', for letting their dogs run the neighborhood rather than being controlled/confined.
When your own dog gets into livestock and mauls them, it is your own fault - not the dog's, not anybody else's. That is what is most disgusting about the Kristi Noem story. (Also I would argue she should have dispatched the dog with a clean rifle shot to the head and not used a shotgun, as the potential cruelty of that method was evinced by what she did to the poor goat.)
In general I don't have an issue with animals being dispatched by a clean shot to the head rather than calling out the vet - in rural America it can be a lot less cruel to do it quickly with a rifle than let the animal suffer (if that's the case) while waiting for a veterinarian to be available.
But dispatching a young healthy animal for being a bad hunter and getting into livestock isn't justified - because it's the owner's fault that both of those conditions occurred, not the dog's.