At the college level, private schools certainly do cater to the individual. I was fast in math, regular speed in everything else. They put me into grad level math and physics classes, while my writing/history/etc classes were still at the normal pace.
I don't see much reason why the incentives would be different below the college level.
Whenever you want to ask "what would a privatized primary education system do", you can often ask instead "what does the privatized secondary education system currently do?"
Except that this is _exactly_ what happens at public colleges as well. In fact, my public high school saw that I was fast at math and got me into college classes early, so it even occurs there to some degree.
Most colleges are competing for students; there are market forces. Public schools have a much stronger monopoly due to the difficulty changing location. Additionally public schools still cost money, if it was useless people would not use it.
By "privatized", I meant a system comparable to the one most education privatization activists push - public money follows the student to the school of their choice, which may be public or private.
I don't see much reason why the incentives would be different below the college level.
Whenever you want to ask "what would a privatized primary education system do", you can often ask instead "what does the privatized secondary education system currently do?"