You didnt mention what position villain is in when they called you as that is important to determine how wide their range is. If villain is in the BB then they could be somewhat wide.
On the flop on this all one suit board with connectivity and villain leads out I would not put them on a flush and I personally even think just draws dont do this much either although its possible. I would be thinking this is a one pair hand that does not want a check behind on this flop to go to a turn. I think most likely we are dealing with Qx or 9x. I think your raise has merits and drawbacks but in the end I am fine with it since it defines the hand more meaning villain has to show the strength of their hand here and we keep control of the pot.
On the turn no club is good but its another card that connects the board. We get checked to and I like checking back here although it looks weird to raise flop and then check turn but it does control the pot now and gets us to a river. This is not a fun board for AA so I am not looking to score a big win here, I am more looking to just score any type of win and get this hand over with now at this point.
River is a bad card other than its not a club. We get checked to and now we can really try to range villain here. I still think they are (were) on one pair hand on the flop. I thought Qx and if thats the case, how many Qxs are calling our raise? I doubt too many Q6, Q4 ish type
hands are calling pre to our raise. That leaves Q8 or Q9+ in the Qx range. All those beat us now except KQ and I am discounting AQ due to blockers and lack of pre 3 bet. If villain had 9x on the flop then what 9x would be calling a raise? Those would be connected 9x like 97,98,109 and J9 with also throwing in there A9 but we block that heavily. All those 9x hands beat us except the rare A9. So when villain checks this river I am very happy and just checking back. I dont think anything worse that could have got here calls a bet if we were to here outside of KQ which is too narrow a target.
