This is the downside of having Green and Pierce on the floor together with one of them as the nominal 2. The correct adjustment may be to have Green coming off the bench again because I don't think benching Bradley in favor of a better ball-handling guard will solve the problem.
Why wouldn't it? Bradley's not even the point guard. He brings the ball up and then passes it off. He's sometimes such an incompetent passer that he can't even execute that correctly. The two post feeds he messed up in game 1 were as standard as you can get. There are two ways to fix this. One option is to let Pierce bring it up all the time but this won't happen because they don't want to tire him out. Pierce only starts to do this later in games.
Option two is have another guard on the floor besides Avery who can bring the ball up, anything to get the ball out of his hands. Option two is flawed too because of Doc. When Lee and Bradley played together, Bradley handled the ball far more for some reason. It was to the point that Lee was significantly marginalized in the offense as a starter. I think they wanted to develop Bradley so they let him handle the ball a lot. It's understandable in the regular season but I'd hope they wouldn't allow it in the playoffs.
Edit: Last season, when Rondo and Ray were out and Bradley started for the first time along with E'twaun Moore, Bradley almost never handled the ball. It was mostly Pierce with Moore bringing it up early in games so as not to fatigue Pierce. Bradley's handle and passing ability have not gotten better since he first started last season. I don't understand why his responsibilities have gone up.