Most states that recognize the coyote issue have year round open season on them. KY has just legalized night hunting for them. The problem is they are making themselves at home in more suburban settings where hunting is rarely allowed. Would you want someone hunting/shooting around your church or parsonage? As a result, the only personnel legally able to "take" them is LEO and Fish & Game folks. I imagine some trapping could be done, but with the domestic animals running around, all it would take is for someone's fido to get trapped and a law suit to follow, and then right back to square one.
I don't really think there are any true predators of coyotes except man. Wolves will kill them if they catch them, but don't prey on them. Bears as a general rule will steal a coyotes kill, but could care less about the coyote. The same for mountain lions.
I hate the thought of a dog being taken by coyotes, but strays are strays... A stray dog can be just as dangerous to my livestock as a coyote. If a coyote or pack of them were attacking my dog, I'd be out there, gun blazing. That's why I got a LGD, and intend to get another.