I've known a few farmers that have shot their own dogs, usually a young dog from a litter of an established farm dog, because they killed livestock. I know they found it a very hard thing to do.
In many countries if a dog strays on to your proeprty and chases livestock or kills livestock the law states you may shoot that dog.
I know from some similar dog kills chicken threads here on BYC that in some states you can shoot/kill a dog that pesters your live stock. There is even a abreviation of SSS (shoot shovel and shut up) I think that is recommended by others on this forum. Many of these people have dogs.
The main difference here is that the owner of the dog is also the owner of the chickens; this doesn't make the principle wrong, or the law a fool.
I know you're a dog lover and I can understand why you might find my view unacceptable but it is not just my view, it is the view a many farmers and smallholders that have to deal with untrained/unsuitable dogs killing their livestock.
I don't have a problem with people shooting a dog that is killing their livestock. Multiple times over the years we've had dogs get in and harass our horses and steers, so I know exactly how that is. I've always been able to locate the owners and firmly let them know that if the dog(s) come back they will be shot. I guess we've just been lucky, because they were never to be seen again.
To me, your own dog going after livestock is a different kettle of fish, because just as you have a responsibility to take care of and protect your livestock, you also have a responsibility to take care of and train your dog. If a person doesn't have the ability to take care of an animal, any animal, properly then they should not have them.
My point was one shouldn't expect to keep a dog, a predatory animal, and chickens, prey animals, together unless they have a plan to either train or contain them. IMO if your dog kills your chicken then that's on you because somehow you've mismanaged your situation.
For those that don't have the time or ability to train a dog, or just don't feel like dealing with it, a simple sturdy and/or electric fence will quickly and easily solve the problem. As would only having either chickens or dogs, IMO when you choose to have both, you are also choosing to take on that extra responsibility.