3/6 turn decision, fire again or take the free card?

Even though the Hero is in a better position, the fact that the BB checked the turn, doesn't this give the Hero a free card also? Given that he has missed both A and J and his OESD hasn't been completed, we can get a free card to the river hoping for an A, J, or K, even 8, to complete (and then safety in calling down). I know we have a range to put the villain on, but assuming he's calling down on a low pair on the board, or even a PP, we can save a bet to the river. If the villain checks the river and we don't complete, we can check again to a free showdown, thus, saving 2 BB. If villain bets, we can fold putting him on a least a pair when we have nothing. If we complete the straight, and villain checks, we bet, therefore, making him make a decision. If villain bets the river, we raise, trapping him.