My buddy and I have used leg hold and snares for trapping. We hunt them with rifle and bow. The adjacent county has had bounty as long as I can remember. Hasn't made a dent on the population despite some making a living that way. But If people spent as much time improving their bedding and fawning areas, they would have less reason to worry about predators of any type. We have had here coyotes for 33 years that I am aware . Do they take fawns? Yes, they are opportunistic predators. So do my bear and bobcat and bald eagle and I bet even my red tail hawks. So do I kill every predator that might take one of my precious deer? Waste of time. Predators exist because of the availability of prey and their numbers will fluctuate. Their is no simpler cause and effect in nature than the prey/predator relationship. Follows such a simple S curve. Coyotes do take deer but more so the mice and snakes and ground hogs and other varmins. They put a hurt on rabbit and quail and maybe turkey. But they are not a deer killing machine sometimes made out to be. I watched this last year as my dominant yote passed thru a prime deer bedding area. As soon as he had passed, nearly a dozen deer, of all ages, up and downwind of where he had passed, stood up and began leisurely feeding in the area. Kill your predators, and if food is available , more will move in and the resultant struggle to establish the new pecking order within the new groups will cause more damage to the prey than an established one.
Improve your bedding and fawning area, control deer overpopulation of holding capacity if needed, kill a few predators when you have the chance, and your worries will be simplified. Love my deer but I can get a lot of enjoyment from watching a coyote do his jump in the air to dive nose first into a foot of snow retrieving a mouse. No different than watching my hawks dive 100+ mph to nail a rabbit.