everyone lie about their hand all the time…
You can play it this way if you think this villain has AK, AQ, AJ in their cold call range pre because they’re not folding AK, AQ, or AJ most of the time. Or You only check if you think he’s gonna stab a lot and you gonna call then check/raise the river.
I like bet/bet/bet.
What do you have when you check/raise/re-raise the turn?
I don’t think you’d have A5 suited and this is why betting here is better than checking. You are opening early position, you have all the Ax suited plus big aces, all of which you’ll keep betting with. You’re gonna check/fold or check/call one time the turn with the rest of your range. When you check raise the turn you only have boats plus AK.
I know you feel robbed that he hero folded here, but from his perspective, he’s beat most of the time and it’s hard for him to boat up vs sets.