It's been a while since I learned the history, but if I remecmber right the first schools in the US were religious in nature. But public schooling was generally a huge equalizer, and made the most advances along with workers rights movements, etc.

That said, there's plenty to be upset about class-wise, just not the class size thing. It's true that rich families have always done what they could do to get their kids ahead, generally with private school and tutoring. They have a much higher odds of getting into the better colleges, and the more elite schools tend to lead to higher pay after graduation. They're also doing everything they can to gut public education, which is the whole point of the push for vouchers (which was especially big during the Trump administration).

There's a thousand more reasons to be pissed off at the rich regarding education, but if I wanted to get into every single one I'd still be in academia (My PhD in Ed was all about that). Actually, now that I think of it, take a look at Learning to Labour by Willis, as I think it reflects your train of thought.