I live in a rural area and we have coyotes along with mountain lions and bobcats. I rescued a young female coyote that was hit by a car. Her hip did not heal very well and I went through the permit process and she now lives with us and all my dogs and a cat, some birds like crows that come to visit and a turkey vulture. I'm also permitted to rescue and care for other wild animals. We have a zoo of the despicable(s). She, Dolly the coyote and my Australian Cattle, Toby eat mice, rats, and gophers, sometimes rabbits and chase off squirrels. Evidently, squirrels don't taste good, something I have not confirmed but believe it just the same. Toby brings many these rodents to Dolly because she does not get around very well. When Dolly does go hunting she prefers gophers because she only has to dig them out and not chase them.
About 30 years ago there was a big effort to wipe out coyotes and they did a good job. They actually paid a bounty and varmint hunters were more than happy to participate. Problems? Wounded and dying coyotes that us locals had to go out, track down with their dogs and shoot them. Evidently, many varmint shooters sit in one place with a high powered rifle and take pot shots at their target and most often don't humanely finish the job if it's not a one-shot kill. Poison and traps got more domestic dogs and cats than coyotes. Also, rats, mice, squirrels, and rabbits overran the area over the next couple of years. The eradication program was never repeated.
My opinion about coyotes if very prejudiced because of Dolly and a life spent living here. She has a very strong personality but is also very affectionate. A few months ago she brought me my very own dead gopher. She dropped it in my lap while I was sitting at my desk. Her face, head, and front legs were covered in grass and dirt and she was panting. How could one turn down a gift from the heart?
I don't believe in wiping out coyotes and I think if a solution is needed that it's going to be difficult and time-consuming and very unpopular no matter what it might be.