Good 1/2 game.
SB (Villain): fishy, on the passive side. Has called off 60BB with QJ on J-7-4-6 turn vs c/r from quiet player. Still, not totally braindead. Have seen him check down turn and river in position. ($120)

BTN: seems solid, hasn't tried any moves. In one hand where he raised pf with KK and c-bet into A-high flop, he check/folded river vs me. ($240)

UTG+1: wildcard. Loose preflop, has donked a few flops for 1/4 pot and followed up with stronger bet on the turn. His flop check is telling. ($200)

Hero covers.

UTG+1 limps, I raise red KK to $12. BTN, SB, UTG+1 call.
Flop ($45): A T 9
Checked to me, I bet $40, BTN folds, SB hems and haws and calls, UTG+1 folds.

Turn ($125): 5
check, check

River ($125): J
Villain bets $35, in a way that chips fall forward

Crying call, imo. He has a little JQ in his range, though it is mostly toppest pair.

I think this is a blocker bet with Jx and Tx enough of the time for a call to be +EV. Probably not by much though.

Why not barrel turn to get value from str8 draws with the potential to get top pair to fold ? By checking back you basically abounce to a guy heming and hawing that hey your A6ss is good. As others have said this could also be a JQ thinking he's good but I find most people will check call that rather than lead out. It's top pair imo but probably have to call just cause it could be a lot of hands. KK on this board isn't turning your hand into a bluff catcher it's about getting value from draws and trying to make villain fold a weak A.

Fishy 1/2 Vs don't fold Arag on an ace high board basically ever.

Why are we betting this flop and why are we potting it? We block the nut SD. What worse hands are we hoping to get called by?

Based on how you described the villain's motion when betting the river I...

A river bet from a passive fish is almost always a strong hand. On the few times he's bluffing, it would be a larger bet around pot sized. This bet screams "call me" which means V thinks he has best hand and can definitely beat Ax. I hate doing it and rarely actually fold KK here, but it saves me money when I do. Fold.

Straight draws and Tx. But I agree potting the flop is the mistake here. We can check and pot control, and perhaps get one street of value on the turn.