I hunt just about everything (aside from birds) with an AR of some sort.
When it comes to coyote I use an AR chambered in .223. I had an AR chambered in .204Ruger and that thing was AMAZING on coyote out to 500 yards but I was offered a ridiculous sum of money for it. With that money I was able to buy 2 complete AR rifles (in the current market's prices) as well as a few hundred rounds of Federal American Eagle .223 (I needed more casings anyway).
More often than not when I shoot coyotes they're on the run crossing my field of fire and at least 200 yards away. Yes I have successfully taken them at those distances - in fact my longest hit was well past 400 yards (around 425 according to my range finder - the old kind that you had to dial in the two images till they made one image then look at the dial for the distance) and when I did hit the coyote went down where it was hit. In fact it flipped over upon impact and stayed down. I've got a friend who successfully killed a coyote beyond 500yds (well ok so that's what he told me but I've no reason to not believe him) with his AR chambered in .223 as well.
Having a high cap mag helps a lot when shooting at small targets running full tilt 200+ yards away from you. Sure I know some guys who have taken coyote with bolt actions, shotguns etc. I've even shot a coyote with a Marlin .44mag lever gun and open sights (distance of about 30 yards out of the back window of my buddy's house) and man did that make a mess!
Even in today's market you could build yourself an AR for about $1000. You just have to hunt around a bit for the deals on parts. If you're looking for a built AR rifle you'll have to be willing to spend at least $1500 nowadays. Maybe in the future the prices will drop down to something more like BEFORE the Newtown incident. Back then I could build a whole rifle for $600.
This is who we are, what we do.