The "prevent self shadow" options works fine, you can verify it disabling all shadows from rendering and the result doesn't change. The problem is that the extra planes have different illumination, creating an strange effect.
Unfortunately, there is not a perfect solution to use billboards in animations or overhead views. For these type of scenes, perhaps you must change to 3D trees. With Forest you can scatter and render thousands of high-poly trees just as easily as billboards.