This would help parents of black and brown kids as well as rich white kids.  Parents would have the same right to shop for schools that everyone else does.  In larger metro areas no one would be "assigned" to a neighborhood school.  Such a thing wouldn't exist.  There would still be a public school system but it would be based on a lottery rather than where people live.  

This would help promote integration and encourage families to move back to urban areas.  One reason why people move to suburbs is because of schools.  Also, many neighborhoods in cities like Chicago are segregated, so the schools remain segregation.  School choice promotes integration without angering middle class white people.  Rather than forced busing to a bad school in the inner city, parents are choosing to send their kids to a specific fancy language academy or baccelaureate program.  Lots of cities under court ordered desegregation have chosen this option.