Coyotes get blamed for lots of things. And it has gone on for years. Like any potential nuisance animal, sometimes it can pay to remove one or two. More on that in a minute.
We have predators, because we have prey. It is usually a self limiting dynamic. The coyotes relationship with deer is to typically take the very young or the very old. Or, the very compromised- deep snow,etc.
Deer populations can fluctuate for a lot of reasons that have little to do with coyotes. Changing habitat, loss of habitat, over harvesting, etc.
When some hunters come up empty or do not see enough deer, they immediately blame the closest animal with binocular vision as the culprit.
Man has tried to eliminate the coyote for over 100 years. The result has been more coyotes. The trouble with removal is that it is not a question of just killing coyotes indiscriminately. It is rather, which one(s)? It is often the coyote that we don't see that is the one that may need a bullet.
Most of think that if we can just kill enough of them, things will be better, the way we want, the way they should be,etc.
I think if someone wants to hunt predators for the awesomeness of it all, I support that. But, anyone that thinks they will improve things by declaring war on the species, you might as well stay home.
As soon as you kill an alpha female it will signal all subordinate coyotes in the area that it is time to get pregnant.
If you think you're hearing a pack of 6 or 8 it is most likely 1 or 2 They produce a multi-timbral call just for that reason.
It is easier to list what a coyote won't eat rather then what it will. And most of that is vegetation and small rodents of one kind or another.
If you get stalked by a coyote during spring turkey season, just stand up or move your gun. They will bug out and most of the time you haven't screwed your set-up. You can shoot if you want to- (see above)
Would I ever tip a coyote over? Never without what I considered a good reason. Not just 'cause they are there. Ironically, I think most of the people that want to form a posse could not, if they thought about it, ever imagine a world without coyotes.