Many intelligent people aren't given the chance to be successful, while others are given the chance, but the decisions they make hinder them from being successful.
I'm a bit biased.
My dad wasn't ever given the chance to be successful. In his version of year 11, he was forced to quit school by the government, and move to the countryside, and he was to stay there for many years. He was doing really well at school...but all that didn't count anymore. After that, he missed out on uni. His younger brother was given the chance to go to university, as by then a new government was in control. Lucky for him he now has a professional job and can send his child to any private school.
I know that nowadays similar circumstances are rare in Australia, but I don't think that it's fair to say that "people in a position to send their children to private schools" are "likely" to be successful and intelligent.