22, and small pocket pairs are blind stealing hands (like QJ type hands) and differ greatly from trash hands like 83 and Qx. Sure with 83 your attempting to steal the blinds, but blind stealing hands also afford you the opportunity to have the best hand if we go to the flop. In addition, we want hands that have some showdown value on their own merit as pocket pairs do (22 beats AK on the river if AK missed, 83 and Qx do not). I know we are essentially flipping by the time we get to showdown if that is the case, but flipping still means we have roughly a 50% chance at showdown if we do not hit our set, whereas 83 and Q7 do not have that equity, especially when they miss.
In your last response, you still discount the fact that we can hit a set with pocket pairs. Trash hands like you mentioned do not do this. As a matter of fact, playing a hand like Q7 post flop is not very comforting either. If we hit our Q, we can be out kicked and potentially lose a huge chunk of our stack. And I am usually not that happy to hit a 7 either, since it is usually not top pair, and if it is, we can be in trouble if we are called preflop with an over pair (88+). On the other hand, small pocket pairs are pretty easy to play. While we are not longer set mining at this stage of the game, when we are in late position, I am happy to open with my pocket pairs. Again, obviously it is nice if we take down the blinds. If we get to a flop and hit our hand, now we are just trying to get max value (12% of time on flop, 8% on turn and river, again, this cannot be disregarded). If we miss (ie. don't hit our set), we make a continuation bet for about 1/3 to 1/2 pot. If called, then we have to re-evaluate on the turn. If we get checked back to, we may look to put in another 1/3 sized bet. If called and checked to on river, we can check down and we do have showdown value if we were getting called down by a draw or high cards (where 83 and Q7 do not have showdown value in this situation). We may also have to check the turn as well for some pot control depending on board texture and of course our opponent. If at any point we meet some resistance- such as getting check raised or our opponent leading out on the turn, then we can dump our hand. No big deal. The beauty is, we should not be losing a lot on these hands either. We try to get it all in somehow if we hit our set, otherwise we try to keep the pot small and take it down on the flop or get to showdown.
So there are several reasons why I am playing 22 and other small pockets here in late position rather than Q7 and 83.