This would only happen if you set the threshold too high. If Harvard wants to be an institution with lots of AA, NA and LA students then just set the threshold at a level where these groups can enter the lottery.
A lottery at any threshold will create a student body biased away from blacks/natives/latinos, since the entire bell curve of their SAT results is to the left of whites/south Asians/east Asians. (Unless the top students stop applying, which is an unacceptable outcome for any elite school.)
[1] https://i0.wp.com/www.brookings.edu/wp-content/uploads/2017/...