Sure, it might stay centered on the star if it started centered. Any mild disturbances would cause the closer parts of the ring to receive more solar wind and

Sure, it might stay centered on the star if it started centered. Any mild disturbances would cause the closer parts of the ring to receive more solar wind and light pressure while the more distant parts of the ring would receive less. However I expect that the momentum in the ring is many orders of magnitude higher than the light pressure, so it might well just destabilize anyways. However it's a 3D system, what if the plane of the ring didn't perfectly bisect the center of gravity of the star? The feedback loop would cause things to get worse, since there's no counter pressure.