The puppy was being trained for hunting. Meaning she killed her for doing exactly what she was being trained to do. Of course she wouldn’t know there are some animals she shouldn’t chase.
Oh, and this is not a one-off thing from Noem either. She also mentions shooting and killing a male goat for “chasing her kids”.
In any case, I worry when someone’s solution to completely fixable issues with other living beings is to just kill them.