First, plenty of people succeed despite public school, not because of it.
Second, how do you define success? If you define success in terms of money, education and intelligence play a very, very small role. The correlation of financial success to who you know and who you are related to is orders of magnitude higher than how smart you are or how much knowledge you have. The vast majority of PHDs are much more intelligent and educated that the vast majority of millionaires.
Schools are not all the same, you know? The only thing you can say without diving into the statistics is that it depends on when and where you went.