The problem is student outcomes aren't linked to how much money and resources the school has, they're linked to how much money and resources the parents (and the network of the parents) of the children have. Networking is the primary benefit of top tier schools, and so unfairly giving a chance to those without the network is the only way for them to move up the ladder as a group.
This will necessarily displace individually who might deserve to be there more, but the problem is an intractable one due to the outsize gains of being part of that network.
I am not endorsing affirmative action, or discrimination against certain "successful" minorities, simply offering a chain of reasoning for it.