Seems to be a bit of a ******* match here folks. Yes, the .223 has its limitations, but under 400 yards it is a better option (even in an AR).
Some people like to shoot lead at eleventy billion feet per second and have a straight line from bore to target at 600 yards, and good for them. But that is not the end all - be all of shooting.
I believe you mentioned 300 yards-ish for your rifle. .223 will be an excellent choice for you, and here is why. (we will skip the aforemented "cheap" arguement as both sides seem to agree on that front).
Ammo is available for this. Check out: Ammunition - .223/5.56
You can find pretty much everything from 55gr FMJ to 62gr soft points to 69gr HP match ammo to just about everything else. So ammo is available.
Barrel life is longer in a .223 than in a 22-250. (Truthfully, I doubt you will shoot enough to really burn out a barrel unless you are shooting 1000 rounds a week, even in 22-250)
A .223 will blow up small varmints (rabbits and p-dogs) and kill coyotes up to AT LEAST 300 yards. I am assuming this is somewhat of a learning experience for you, so lets keep things reasonable.
No matter what you go with, take care of your rifle and it will take care of you.
I hope that this info was somewhat helpful for you. Yes, there are other cartridges, yes there are better cartridges other than .223, but no matter what you get, learn to shoot it, not just blow through ammo.