The obvious counter example is the US, where public schooling is sectioned off by neighborhood and rich kids still attend school together because they can afford to live in their respective neighborhoods. Even worse, the US spends more money per child on public teaching, and still has some dreadfully bad public schools due to various inefficiencies.
In short, I think your assessment is backwards private college/lawyers/your-favourite-resource don't favor the rich, instead the rich build private college/lawyers/your-favourite-resource to favor themselves (or their children).