Actually, the main reason coyotes spread out of the southwestern part of North America into other areas is because the wolves were eradicated in those other areas; coyotes simply moved in to assume that predatory role.
Farmers/ranchers have historically had a very antagonistic view of coyotes, and other predators, because those animals often posed a very direct threat to their livestock (and financial welfare). They still do to this day, but our views and techniques for wildlife management have evolved over the years.
Honestly, killing a few coyotes (whether it be as part of a humane effort, population culling, or general sport hunting) doesn't have much affect on the overall population. Coyotes are doing fine in North America; hunting and trapping efforts don't pose a threat.
They not just doing fine, they be thrivin LOL
I mean if ya just want to kill something, go for it. But it may well survive.