Hi Mickey,
Let me answer "metagame" first before math, IMO no this is not bad for metagame because my stealing range is very wide, and I will often have nothing that can win the pot unimproved in spots like this, so I will sometimes be checking behind turn and giving up if I don't hit the river. For that reason the fact that I am sometimes calling a river bet unimproved is a fine balance. The only thing I don't balance in this spot is hands that check behind turn and then raise river unimproved, because I think just betting twice is the safer, better option.
Now to math, this will get nasty fast so I'll try to just do a bit of an overview. I don't remember the exact action so we'll go with your $40 in the pot number. So there are two cases to start off with, the times my AQ is the best hand on the turn, and the times it isn't. I am/was assuming he would never fold a better hand when I played it, if you disagree with that let's talk about it. I think the typical villain online these days is never folding a pair there (nor should he IMO). So the times I'm behind there is no argument, we want the free card, we are a dog to win the hand.
Now, the times we are ahead is more interesting. When ahead, there are a few things that can happen if I check. 1) we can fall behind when he hits the river, and it either goes check/check, or he bets and we call and I lose. 2) We can both improve and it goes check, bet, call, or bet, raise, call. 3) Nothing changes and it goes check/check, or he bluffs and it goes bet, call and I win. 4) I improve and he doesn't it goes check, bet, fold, or rarely check, bet, call when I was ahead already (he looks us up with worse ace high maybe), OR he bets as a bluff and I raise and he folds.
Case 1) The board in question was something like J236 with two diamonds I think. Best hand that was already behind he can have here has 9 outs (say K4 of diamonds which is the best he can have, other hands have fewer outs because the diamond cards that pair them are not outs). So we have 8 cards accounted for, 9/44 times he improves on river, 35/44 times he doesn't. So right around 20% he improves. So 20% of the time I lose $50, the $40 in the pot and a $10 river bet. If he bluffs the other 80% of the time, my net EV is -$2. So that's not great, but still, close to even, and I believe I gave horrible conditions for myself and still come out ok. Now its even worse if he never bluffs, so against a passive non-bluffy opponent yes I'd err more on the side of betting the turn and taking a free showdown.
This is already getting long so I'll say case 2) Is all good for me, basically depending on his hand there are between 2 and 7 "outs" that improve us both and make him lose more. 3) was counted for at the end of 1). 4) is all good for me too. So 1) and 3) I'm a slight dog under terrible conditions and 2) and 4) are extra +EV for me guaranteed, so I think this is pretty clearly +EV for me overall. Sorry for the hand waving but you can add conditions and calculate math of these things forever, I just wanted to get a general overview of the factors at work here.
Basically we are just disagreeing on assumptions probably. If he never bluffs river, you are right. Perhaps more importantly if the pot were larger, that is a huge factor in the above math. If the pot were $100, I'd certainly bet to protect my hand. Though note if the pot were $100 for whatever reason, its a lot less likely my AQ high is good!
-DeathDonkey