I prefer to lead out on the flop. We are likely ahead, we buy up our equity for over cards, we sometimes get floated by broadway type hands, and we have a stronger range in general.
As played, the check-call isn't terrible. Button will often try to steal the pot or value bet smaller pocket pairs.
Now, the ace on the turn. I think that's a good card to show aggressive action on. You can donk, check call, or check raise, and all will be effective for making a move on the river. The only hangup is if the Ace pairs.
So here's the question: what are you folding out on the river exactly? Is 9s full going to fold to a single bet? No. Is any Ace ever folding here? Clearly not. Maybe we fold pocket 8s? I don't think TT+ is ever folding to a single bet here. Everything else that folds is stuff we already beat.
I actually think this is a thin value bet, and you are looking to get called by pocket pairs. 22, 44-66.