A dog that is aggressive toward family members and has gone on a chicken-killing rampage after showing itself untrainable is NOT a re-homing candidate. A billy goat that attacks is also a dangerous animal.
When you have a proven dangerous animal, you deal with it humanely and quickly.
Correct.
I absolutely concur that the dog should be shot. Also it is suggested that it is a registered breed, so it was expensive, so shooting it was not done without reflection. There are dogs that are unstable and they must be done away with.
Some have mentioned "Re-homing" the dog. Why? There are many many nice shelter dogs that waiting for a new home.
OBTW, there seems to be an inference that hunting dogs need to be pure breed to do well hunting. That is not true. When I was growing up, the best hunting dogs were mongrels that had the right instincts and the desire. Just recently, I owned a pure mongrel (We called it "the brown dog.") that was as fast as lighting and instinctively a pointer. That dog was so fast it once caught a bird in mid-flight and used to kill a couple of squirrels every year. That dog would be doing a full out sprint and lock up and point at a bird. The dog's point was the classic one foot up, tail straight out and head pointed at the point of interest. Unfortunately it was extremely skittish. If I ever had shot a shotgun around it, I would have spent the rest of the day trying to find it.