It's always interesting to see the "anti" tag come out whenever information that doesn't meet some long accepted local wisdom comes along.
Studies--by game biologists who are hunters, funded by hunting and fishing fees and taxes on sporting arms and ammo--have demonstrated the value of coyotes a number of times.
In some ways it's simple, in others a bit more complex.
Coyotes are primarily mousers and bunny chasers in most of pheasant country. They will take other game as the opportunity comes along, but unlike fox, skunks, and coons, they are not particularly focused on nests and birds on nests.
Fox--which have been studied heavily-are very adapt at searching for and finding birds on nests. At night they will kill the hen and eat the eggs as well.
Coyotes are hard on some other predators, particularly fox. Their home range is larger than fox--take a given large farm with habitat here and there--it can support more fox than coyotes.
So--if you set out to kill all the coyotes, the fox that come in will kill more game birds than when you weren't killing anything!
Of course, it's really hard to kill off all coyotes.
But wait there, here comes another problem. Established pairs or packs of any predator tend to limit others of their same species in that area. As you kill them off, you open the door for more travelers to come in, and all of you sudden you have more predators there than when you started killing some.
If your head is spinning, there is one incredibly simple solution--habitat! You do it right once and it's effects last a long time with only occasional maintenance.