My first dedicated coyote rig years ago (well, second, my first was a Marlin 60 .22lr) was a Rem 700 heavy barreled .22-250. I bought that because everyone had me convinced that the .22-250 could kill a dog as far as you could hit him, and a .223rem was only good within 200-300yrds.
And then I eventually bought an AR-15, NOT for coyote hunting, but I took it out a few times and realized that I wasn't seeing any practical difference in the two in the field. Assuming you can put lead to fur in the right place, any dog within 500-600yrds is dead with either one of them. Sure, the 250 has 50-100yrds advantage on energy and trajectory, but again, either one is groovy to 500-600yrds.
HOWEVER, for what it's worth, that's considering similar barrel lengths. There IS a big big difference between a .22-250 and a 16" AR-15 carbine, heck, there's a big difference between a 24" AR-15 rifle and a 16" AR-15 Carbine. The Long barrel AR's can push 50 and 55grn pills to 3200-3400fps, 16" carbines are lucky to top 3000, often falling into the 2850-2900fps ballpark.
Personally, I have a 6.8 SPC, getting a .243 WSSM from Mike, a half dozen .223rem AR's, and 2 boltguns in .223rem for coyotes. The short barreled AR's mean shorter shots, but the long ones give plenty of range even out in the wide open spaces of western KS.
As far as a factory rig goes, a Bushmaster Varminter Special for an AR rig, or a Savage 12 Varminter Low Profile (12 VLP-DBM) for a bolt gun, either in .223 or .22-250 would be my picks. Topped with a 4.5-14x50, 6.5-20x50mm or 5.5-22x56mm glass, those would be my set ups if I were picking ONE to start over with.