[mjkidd]

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Obviously bet/fold is exploitable. But so is check/call and bet/call and check/fold, if we always do it in this spot.

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Every action is exploitable.But if villain would play his hand perfectly bet/folding and check/folding is by far the most costly ones.
If villain plays perfectly he knows that your range is 66-AA.He knows that you´re going to bet/fold river with all non-set hands.I don´t expect anyone to play perfectly but if he does it is a 5/9th pot mistake.

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Villain is unknown, presumably so are we. Theres no way he knows that we're going to bet/fold JJ here. Do you really expect an unknown to bet/fold JJ in this spot?