Someone I know is an illegal immigrant from Central America. She cleans houses for a living. Her husband is also illegal immigrant and he paints houses. Her daughter is going to one of the top private schools in SF, on a full scholarship. She dreams of Yale, and I would bet that she gets it easily.
If you are dedicated to education, there is plenty of opportunity to succeed. It’s sad that some bloodlines don’t make it out of poverty easily, but that’s just how the world works. For those that make the effort, it’s very available.