The strangling part may have been cruel, but was accidental.
Cutting the throat is the kosher method of slaughtering all food animals and is done to prevent suffering.
Where is the firing offense?
Cutting the throat of a dog whose neck is being compressed in a noose, under a very high stress situation is not equivalent to a quadruped being calmly led into a kill box and having its throat slit. That dog died in extreme stress and fear. It could have been handled much better. From what I understand, the woman grabbed the dog’s collar and triggered the fear/defensive reaction on its part. That cop was a fool and has no business in the cop business.