Easy call?

1/2 NL. I have 260, BB has 220, UTG+1 has 180, MP has ~150.

UTG+1 opens for 6, one caller, I call with 6/6, BB re-raises to 10. All of us call and 4 to the flop.

Flop: 6/8/2 rainbow
UTG+1 checks. MP checks, I check, BB pots it. UTG+1 folds, MP folds, and I call.

Turn: 6/8/2/A rainbow
I check, BB checks.
Is this horrible? My thinking was that the Ace scares JJ/QQ/KK and a bet from me might fold out a hand that will call down a moderate bet on the river. Also it allows Ax to bet and gives me the potential to extract more value

River: 6/8/2/A/2
I bet half the pot. BB pushes all in. I call.

BB had AA. Anyway, I'm not trying to be results oriented, but is there ever a time when this is a fold? I'm presuming you get stacked here every time and are happy to get all your money in because of the times villain makes this play with a big ace?

Also this hand got me thinking about the validity of pushing the river against players capable of making a big call with something other than an ace here (like KK). How often would I need to be called there to make it profitable presuming villain always calls a smaller bet on the river, but won't call the push every time?