Still not sold on the x/jam, lets think about villains range. There is just very few draws on this board by the turn, for example, maybe only AJ that floated the flop if he floats the flop with this. But then if hes floating AJ, he is floating AQ which makes top pair almost as much as the times he has AJ that we can call "air". When you include two pair with QTs that villain has and atleast TT for a set, he has more value than bluffs.
If we say villain has 66-99 then on the turn this is around 35% of his range which could make a check jam totally cool if you think he is going to bet fold these on the turn, however, i think just betting again guarantees these fold every time and it possibly folds the middle part of villains range that makes up 25% of his range (T9s,KTs,ATs, JTs, JJ) which he would otherwise just check back with on the turn (and if he does call with these we will have a very profitable river shove if we set the stacks up better.)
So yeah i just kinda feel he makes more mistakes when we continue betting, and that hopefully shows how we have more FE when we barrel rather than x/shove. without even considering what happens when
Hero checks, villain bets $3.50, Hero jams, Villain has to call $ 4.15 into a pot of $12.30 or something, only needs to be good 25% of the time, kind of get ourself in trouble if he does ever bet some of his pair hands that arent really low because he might just talk himself into calling off because we can have AK