My son is 3. I've taken him shooting half a dozen times, though he's only shot a 'gun' once - a Walther P22. As in, one bullet. I let him do it so he would know what it was like, as he's totally transfixed with firearms, and I thought he needed an example of what they are capable of.
Yes, we both were wearing eye and ear protection while we did this, and yes, I was holding the gun as well. He only fired three times, and there was only one round in the magazine each time (ie, we shot, and then I put another magazine in with only one round, repeat). It was enough for him for the time being, so we stopped then (and no, I didn't provoke him to try, he asked).
I should note that he's got a full range of plastic and metal toy guns (well, 3 or 4), and he has been taught the 4 rules using them; I require him to, at this age, to at least not point them at other people. He's pretty good at it, and while we shot he did use the sights (he was more attentive and focused than the kid on the video, at least - who I'd not put at 3, personally - not quite. Maybe 2.5 or so. Either that or my son is a bit of a giant).
After that, we went back to the Daisy BB gun, which is more his thing at this age. But I'm not going to begrudge my son an attempt. I mean, c'mon - it's a .22LR. It has barely any recoil whatsoever.