Honestly, I think I play this about the same as you with no read on my opponent.

I don't like shoving the turn, as almost all the hands you beat are folding and none of the hands that beat you are folding.

The hands that he beats us with on the river are AK/TT/QT/QJ (in that order of likelihood). He could could just as easily have any pair, any A-x, or any two cards that he liked enough to call preflop. By checking the turn and the river, we induced a bluff from all of these hands, so we have to call.

I 100% make a note of what this player called with preflop and on the flop here, as well as if he checked behind on the turn with a scared QT or a trapping QJ.