If I may, and please forgive me if this comes out wrong, I would suggest the following:
When keeping smaller groups of livestock, say <50 (chickens, turkeys, sheep) I find it in ones best interest to provide the maximum amount of physical protection from the elements and predators.
I cannot help but to be disappointed in MYSELF whenever a weasel, coyote, fox, owl etc. is successful in removing one or more of my herd/flock. I do not blame them (predators) for they know not what they do. I figure out how they penetrated my holdings and do my best to fix the problem.
What you did for that coyote was the spot on. He had no alternative but to suffer until time, weather, starvation or you, took his llife. I commend your difficult decision.
There needs to be a sort of truce when it comes to creatures of the wild and the creatures we keep and care for. We need to understand that it is our responsibility to protect the domestic creatures from harm they are unaware of. Conversly, it is our responsibility to respect the wild creatures and protect them from themselves, and us by doing whatever possible to discourage them from seeking a free meal on our dime.
It is late, I apologize for the long-winded-ness.
Respectfully,
FireMedic