I think I understand what you mean by "commonly used", as in recognizable to the average hunter? I really would not consider .218 Bee, .17 Rem, or even .22 Hornet to be commonly used in that sense.
Yes, .223/5.56 is rather anemic by deer hunting standards. Most non-gun people have no idea, though, because all they here is "High-powered assault rifle" on the evening news.
I remember seeing a "comic strip" in the paper many years ago around the time of the Clinton AWB. I don't remember the name but the artist was a well-known leftist. It showed a hunter with a deer in his sights, shooting an AR rifle. "Rat-a-tat-tat" (full auto fire). Another hunter stands where the deer was and all he sees is red spatter. The first hunter says "Did I get him?" The other hunter replies "I think I found a piece of antler!"
So yes, they deliberately want ignorant folks to believe that the guns they want to ban are uber-powerful full-automatic death rays. The way to counter this is not to say that the 5.56 round is underpowered and less dangerous, because everyone is painfully aware that an AR can be horribly lethal in the hands of a lunatic.
What I think is helpful to point out is that they are not fully automatic, they are functionally the same as many hunting rifles, and the round they fire IS underpowered by deer hunting standards.
Yes, I know the second amendment is not about hunting, but that's a whole other argument. My point is that if you can show the average Joe who gets everything he knows from soundbites that he is being lied to by the gun ban crowd, maybe he'll be willing to listen a little further instead of just tuning you out as a "gun nut".