1. If you have a good school, what is there to stop rich people from buying up homes at the school - eventually turning it into a rich man area?
2. If you force the good school to take lousy students and the standards drop, teachers, students and parents will just move to the next good one.
In summary, unless you constrain all schools to be the same as the lowest common denominator, some schools will be good, even significantly better than others. And those with the means will gravitate towards them for the sake of their children.
This is a true story (from a trusted source): I know of a boy from a poor family who because he lived nearby and by the luck of the draw managed to get into a fairly good school. Both parents uneducated, working hand to mouth - hence no ability to coach the kid plus no money to spend on tuition. Kid performs poorly in school. Teachers get frustrated that he is "dragging" the class down - piles on him. In order to seek attention and in frustration, kid acts up further - now notorious in class as well as a laggard. A vicious cycle ensues.