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Death and suffering are two different things. I don't kill healthy coyotes that leave my stuff alone, I only kill those that chew/dig into the chicken coop. The number of coyotes I've killed? NONE.
I don't want to see all coyotes dead, they have a valuable role in the ecosystem. But they need to stay in the ecosystem and out of my coop. The OP hasn't had a problem with this coyote taking their pets/lifestock, then why are we suggesting the OP kill it? Because it is suffering, mange is a horrible thing, causing the animals to bite and scratch themselves to death, compromising their immune system making them easier vectors for disease.
Just because we suggest someone to put a coyote obviously suffering terribly with mange out of it's misery, doesn't mean we want coyotes hunted into extinction. What a horrible accusation to pin on people who are suggesting the OP simply humanely put the animal down. What is wrong with that? Humanely killing an animal and allowing one to suffer are two completely different things. Animals that kill my pets and livestock, I may wish gone or dead, but I never wish them suffering or pain.