Nothing wrong with the 223/5.56 for deer. In NM it's legal and although I don't hunt, I have friends that do and it works fine. Most have a twist of 1/9, and the Nosler Partition at 62gr drops them like a sack of potatoes in the neck or behind the shoulder.
My Colt AR has a 1/7 and using any Barnes Bullet ammo or 75gr Parti BTHP would do a great job. Hornady Superformance 75gr BTHP gets you up close to 1500 lbs muzzle energy, and I think it would drop the biggest buck that crosses your sights.
Grizzly is still on the endangered list, so they will only be hunted when they come off.