Idaho has no caliber restrictions either...as long as it is a centerfire. A .17 Hornet is perfectly legal for elk.

As to the 22-250 vs 223, I have killed plenty of deer and antelope with both. Never saw much a difference but I do think speed helps with the .22 cals for bigger critters. The smallest I have killed an elk with is a 6mm, and it worked just as well as anything else.