I think the .223 Rem is a great cartridge for coyotes under 300 yards. However, it's not all about cartridge, you need a scope or at least a red-dot sight on whatever you use.
High-velocity .223 CF rounds, using varmint bullets are safer than WMR or .22LR rounds because they turn to dust on impact and don't tend to ricochet.
My son who lives nearby uses a .30-06 for everything from woodchucks and foxes to coyotes, deer and moose. He's deadly with that rifle!
I'm more of a gun nut than he is, having a .22 WMR CZ, a .223 Tikka, two .243 Win (Tikka and Rem), and a .270 Win. Any might do the job, but I tend to take either .243 when looking for coyotes at ranges up to 350 yards, but have shot them with all my CF cartridge rifles. I've used the .22WMR in a pinch within 100 yards, but don't have confidence with it.
People are like rifles. Some are tried and true, having great eyes, personality, and fun to be with. Others never seem to hit the mark with you. Still others go off half-cocked. Still, it's nice to know most of them.