Actually you haven't. I have read all of your comments here and the only thing you mention is 'being exposed to a vast set of people with different experiences and biases'. You do not list social skills anywhere.
Even if it is true that public school provides this exposure - which is certainly questionable depending on the school - you have nowhere explained how this exposure at school helps people to develop positive social skills.
Even if it is true that public school provides this exposure - which is certainly questionable depending on the school - you have nowhere explained how this exposure at school helps people to develop positive social skills.