I would remind all that game laws are not about what a cartridge can DO, but more about what the bulk of people will do with it. And, what the state considers best for the SPORT, overall.
Also remember what existed, or was expected to exist soon, when the law was written. And what changes have been made since, if any.
Up until fairly recently, .22 centerfire rounds were either loaded with military FMJ (.223) or varmint bullets, very poorly suited to cleanly taking deer. There were a few heavier built bullets, but only a few and those were a handloading proposition. The guys using those weren't the problem, the concern was large numbers of the general public shooting with varmint ammo, resulting in lots of cripples and lost deer.
Generally speaking, you'll want a bullet 60gr or heavier, and one that isn't a match bullet or a varmint bullet. Everybody making things like that has somebody to answer their phone. Call them and ask what the bullet is made for.
Newer rifles with the fast twist are built to give best accuracy with the long heavy bullets, older guns with twist rates for the varmint bullets not so much, but MIGHT be good enough with certain bullets. Only testing by shooting will tell.
One of my rifles with the old slow twist rate will put the 55gr and under bullets into an inch or usually less @100yds. Shooting the Sierra 63gr semi spitzers (the old time "deer bullet") its best is just under 2" @100. Good enough for deer, at reasonable range but not good enough for small varmints with that bullet.