I have never shot a coyote with a 223, but have shot lots with 222 and 222 Rem Mag, 243, 6.5CM, 260 Rem, and 308Win. When the shot is placed correctly, generally a coyote will go down on the spot regardless of the cartridge that did the damage. Gut shot, leg shot, non vital shot, what ever, they all spin, then run--some are recovered with a follow up shot, some are recovered after they bleed out, some are never recovered-which I hate. My go to gun for coyote hunting in the open is my Tikka 243, I use either 58 or 87 VMax, both are absolute hammers on coyotes, provided I make a good shot. If the hunting scenario will be a little closer shooting I will take my 222 Rem and keep the shots inside 200 yards and only on stationary targets. The 222 Rem will kill them DRT way further out, this is just my self imposed limit because I hate chasing a wounded coyote, I have too much respect to take a less than ideal shot.
Availability and variety of ammo is so similar between the 223 and 243 that I would call it a wash, but maybe the 243 has a better variety of dedicated hunting ammo-you will be hard pressed to find FMJ surplus style ammo. The 223 has lots of non-hunting specific ammo which is tempting to use and I would suggest for coyote hunting is not a good choice, but thats just me.