3for3 said
Do real life Villains call here with Ax (She actually had ATo with a back door diamond draw).  I think in general they do not, but perhaps their checking range is so weak that we can bet 2d pair for value.  

Some do, some don’t. Your hand is almost always good, though, so there’s little to lose by betting. And if you don’t bet, you’re getting exploited by all those who do call. If you don’t find the right bluffs here, you’re getting exploited by those who don’t.

I think the problem you’re running into is thinking in terms of the absolute strength of a hand (second pair) instead of in terms of ranges and incentives. Villain should call your small flop bet with loads of unpaired hands. Most of those are still unpaired (Ax, overs and backdoor flush draws, etc) on the river.

Your own range is heavy on unpaired hands as well. You bet lots on the flop (I assume), and you have a lot of incentive to bet pairs on the turn. So when you check back turn, you also have many unpaired hands. That gives Villain incentive to value bet her stronger hands. Any pair can probably be a value bet for her, so while she can and should check some pair or stronger hands on the river, she’s very heavy on unpaired hands. If she folds them all to a bet, then your bluffs would be profitable.

Essentially, both of your ranges are pretty weak because of how little money has gone into the pot so far. Hands don’t have to be high in rank to be worth putting another bet into the pot.