Just checked an article I wrote for GUNS magazine last year on using .22 centerfires for big game. I researched the rules in all 50 states, and they're legal in over 2/3, and legal for some kinds of big game in several of others.
As I noted in the article, quite a few hunters (usually older) have an immediate gut reaction against hunting deer with "twenty-twos," probably due to .22 rimfires. The same hunters are usually fine with using 6mm's, though sometimes (because they're old enough to remember the day before 6mm's) desire a minimum of .25 caliber. Guess what? The difference in diameter between a .224 bullet and 6mm or .25 is about the thickness of human fingernails, though of course depending on individual nails.
I'm "older" and my gut reaction is "why do you need something as big as a 6mm?"
"...the left considers you vermin, and they'll kill you given the chance..." Bristoe
