What you are talking about it school choice, and in many versions of this parents are allowed to have the choice of private, charter, parochial school in addition to district schools.

Milton Friedman, a Nobel Laureate in economics, set up a foundation to advocate for it. I am not sure you are correct that biggest group blocking this is wealthy homeowners.

http://www.edchoice.org/school-choice/school-choice-in-ameri...