I have no need for a .223 varmint rifle....But if I was to build a designated varmint gun, I would use a short-action with a standard (.308) bolt face, and chamber it to .22-250...That way you can use that action for other calibers later on.
However, that is just me....
A .223 varmint gun would be great coyotes and other varmints and for practice. I just don't coyote hunt enough. WhenI do, I usually take my Accumark .257 Wby Mag or my .17 HMR. My buddy I usually go with has a Rem 700 Varmint SF in .17 Remington Fireball, but unfortunately they have discontinued that caliber. It is a sweet shooting little caliber. It hauls ***, with virtually no recoil, uses a .308 bolt face, and a standard short-action receiver. I've tried to buy it off him several times, but he won't sell it... You can still buy one in .22-250, .223, .220 Swift, and .308 Win.
.220 Swift is another awsome varmint caliber. I had one of those for a little while.
Centerfire Rifle - Model 700 Varmint SF - Remington Centerfire Rifles
If you come across a smoking deal on a Howa .223 I wouldn't pass it up either.
My dad bought a Weatherby Vanguard (aka Howa) in .223 about 5 years ago. It has a 20in heavy barrel 1-12tw. He bought it brand new at Sportsman Wharehouse. They were on sale for $339. He didn't really want it at the time but it seemed like a good deal so he bought it with the intentions of re-barreling it to a .223AI later on. It turned out to be such a good shooter he's never done a thing to it except shoot the heck out of it. He did put a Timney trigger in it though. He has a 6.5-06AI built on a Howa that shoots very well too.