That is highly unlikely. Even at elite Ivy universities, a majority of students attended public high schools.
Public education is a lot better than people give it credit for. There's wide variation in quality but there are certainly plenty of excellent public schools available.
Old Greenwich, Cos Cob, and Riverside have great public schools because that's where all the IT professionals, quants, and middle managers move to raise their families. Those districts have access to many adults with an interest and ability to participate in the school community, and so the district ends up with a lot of extra resources.
But the hedge fund upper management send their kids to Brunswick School, Sacred Heart, and the like, and the district on the super rich and the poorer, primarily-Hispanic sides of town, has a poorer reputation.
Based on my experience in high school, the opportunities for a good education were there, but not every student took advantage of them (or was able to take advantage of them). While the public school system has its flaws, the problems with education in the US are multi-factorial. We could do better by having more teachers and funding for schools and modifying our system for educating kids, but we'll hit a ceiling unless we also improve the quality of family life for students at home as well as the US' cultural attitudes towards education.
For example: Stuyvesant is a public high school. But is it actually? There's a 3 hour test you take in order to be admitted. 900 or so kids are admitted. How is this "public?"
There is nothing wrong in segregating kids based on their ability. Public universities use tests to take the best kids and this should happen at all levels. A black kid who is smart should not be held back just because he lives in bad neighborhood. He should get into a better school if he can score more.
Yes. That is how it happens in urban India. Even for nursery admissions schools interview both the kid and parents to see if they have potential.
You are right, he should. But he is statistically very much likely not to based on his upbringing and systemic discrimination.