If it becomes a problem, it needs some control, I agree.
If it attacks poultry or kids, I agree, it needs to be controlled.
But here's a few things to consider: we as humans have been dropping the overall size of megafauna since the wooly mammoth through hunting and development.
The coyote was, at one time, kept in check by cougar's, mountain lions, wolves, etc. We pretty much took care of Grizzly bear's in California, and mountain lion's are less and less spotted. The impact this has had on the local environment is astonishing. Everything in a biological system is linked.
Now there is no salmon. These are truly canaries in the mineshaft for water health.
It works both ways: kill the natural predators, the prey are unchecked. Kill the natural prey, the predators die off.
A few things about the coyote: here in California, about 70% of it's diet is in the rodentia in the ground: gophers, moles, ground squirrels. In fact, coyote urine is sold as a gopher control.
The coyote also has a unique symbiosis with badgers. It is a fascinating sight to see: a badger is a digger, a coyote runs. The badger will out-dig a gopher anyday, but can't chase them on land. So the badger digs and the coyote catches the gopher as it sprints out of the hole. They share the bounty! This symbiosis starts young... I have seen plenty of sign of them both going after gophers and voles in open meadows.
Gophers are already a large pest, but if we had a larger population of Barn Owls and Coyotes in the open areas, their natural predators, gophers would be more in check.
Equilibrium is a hard notion for humans. We think kill them all if they're a pest. Wrong. Keep the numbers healthy. Keep the diversity levels up. Control has come to mean eliminate in the world of integrated pest management.
Another example is the erosion and devastation caused by overpopulation of Elk in Yosemite. The cause was the lack of wolves to control them.
A single species becoming predominant is an indicator of unstable conditions, as in no diversity of plants, prey, etc. Whether ants in your pots, mealy bugs in your crops, or coyotes on your farm, diversity is the key to control. When they start attacking humans, they're hungry. Means they can't find sufficient natural prey. When they start migrating, means they're being pushed out by development or lack of food.
Unfortunately, we're so far down the line, we can't comprehend a diverse system anymore. Our cheap and quick answer, as it has been for some time: wipe 'em out. Then move on to the next in line when it becomes the new pest.... That way insects will inherit the earth!
So, keeping the numbers at bay to protect your animals, kids, etc., I'm all for.
Gunning everyone at sight, not so much....
BTW, Getting a outside guard dog worked with our sheep....