A6s vs x/shove blank river

Honestly villain line looks like a pocket 77s. He check calls flop, turn again he check/calls and lets you get the river as its a blank like you said. And when all the flush draws missed he is check/raising for value as he is ahead of your range for sure! I think you are either way ahead or way behind, I don't see any Ax hand in his range that he would check/shove the river. Also I don't quite understand why are you value betting on the river? If he had a flash draw he will fold the river. I don't see any weaker holding that will call off 3 streets of value bet as you are represeting exactly what you have, top pair. In this case you don't have a kicker by now so A7+ beats you, A5/A4/A2 its a chop. You don't want to bet a pot where best case scenario its a chop, as you will just pay more rake to the site! I would check back this river and take it to the showdown.
Next time at similar spots try to ask your self, do I get calls from worse hands or do I get better hands to fold? In this case answer is no to both questions, as any weaker A is a chop and any stronger Ax you loose to. I don't see any hand that you beat calling 3 streets.

I think you're over-complicating the range vs range. Is your opponent thinking the same?

It's NL2, and you could play the population tendency.

If your opponent did PF call, flop call, turn call, and on the river, check raise all-in, then you should never call with a pair. When you have a set, or lucky straight with a gapper, try it yourself, and see how many people call with a top pair or pocket pairs.

Here is one trick move you could try in NL2. On the turn, spend more time, and then bet 1.2x pot. If opp calls and check on the river, check back. If he bets even 1/3 pot, just fold. You will see many fishes would call 1.2x bet on the turn with A2-A5, or 9X.