I am with most everyone else here. Seems more of a shot placement problem than caliber problem. I would dare say that there are more coyotes killed each year with the 223/556 round than any other round.

Without getting into a huge debate over this following comment, I will also point out that it is currently the round of choice of the US military and military and LE around the world. Although not the most effective combat round and yes there has been a lot of debate about its effectiveness on the battlefield, fact is it does its job decently against 200 lbs targets and that is with military crap ball ammo and not modern BT ammo. I have no doubt it will take down a 25-40 lbs coyote or other predator with even moderately accurate shot placement.

That being said, I currently use an Ar in 6.8 SPC. It's my latest AR build and love the round so far. I would definitely check shot placement before ditching the 223/556 and then if still not satisfied, look at other AR-15 calibers like the 6.8 SPC among others before switching to the heavier, more expensive AR-10 platform.
Thanks and God Bless,