> It's a perfectly rigid and self-sustaining caste system.
The problem with your statement is that all 3 qualities you have listed could actually be acquired at the cost of $0, either by the parents raising their children differently or by the child itself if it desires so.
Therefore I can't agree that it is anything resembling a caste system. (in such systems there'd be use of force for anyone who steps out of line)
The only way that I see how you could argue that these examples prove that it's a caste system is if you denied that there is free will. Yes, if there's no free will then we only react predictably to the environment like mindless machines so being unable to put your children into private schools will cause them to be poor if they don't have some amazing genetical quality that allows them to transcend the competition.
I strongly oppose this line of thinking. Why? Because it leads to Nihilism and a sense that everything (including your own life) is worthless. It would also allow you to do whatever you want without following any moral code if you take this philosophy to its logical extreme.
Edit: What I mean by that is that it's not like parents do not have choices, no matter how poor. They can teach their kids moral values and discipline, they can encourage them to do something useful with their lives. And the child also can do something about its situation once it gets older.
The economist Thomas Sowell made the very strong argument in one of his books that it is actually culture (= set of all social norms and rules) that determines success of a group. For example some Asian cultures or Jews have always been highly successful, no matter where they immigrated to and no matter how much they were persecuted. In case of the Jews they've faced unequal treatment and persecution for thousands of years.
So what makes them able to overcome all odds? They raise their children with certain values (I doubt its the religious values) that make them (on average) outcompete the children of all others.
So it doesn't matter that much if I'm poor, if I'm rich or what skin color I happen to be born with. If I'm white and from the upper class and decide to use the N word in every other sentence, dress like a Gangsta Rapper with pants down to my knees and have no discipline or work ethic then I'm likely going to fail hard in life. If I'm black and decide to learn how to speak eloquently and dress well then I might become the US president.