Frame dragging (or spin of central black hole) would not impact "which plane" in any significant way. "Spin of central black hole" itself is a result of conservation of angular momentum. So, it is possible that rotation of the plane and that of the black hole are in the same direction in high number of galaxies. It is the process of formation (along with conservation of angular momentum) that is supposed to set the direction. Frame dragging follows the direction already set.
Just a thought -
However if there is anything like that, then it has to be that "space itself rotates" (kind of a space hurricane), rather than frame dragging, because frame dragging is a consequence, not a cause. A space hurricane (if possible), can also impart excess gravity and thus can render dark matter unnecessary. The space hurricane would be cold, and transparent in more natural way as compared to mysterious dark mater.