more than public schools. My daughter had been in a famous local private school since she was a second grader. She was very happy and popular there because of her sports talent. She went to a public high school for one year for the 9th grade due to our moving. However, she did not like that school and went to a famous boarding school in east coast area. Even she liked that school very much, she was still unhappy sometimes. She was under some peer presures because she was thought as a genius in math and science and there were too many strong sprorts kids. She could not be popular as well as in other scholls also because there were too many rich kids using money to do some thing to attract others. But then, my daughter always thinks I am her good friend and tell me her thoughts and problem on time, even some thing happened in the midnight by phone. She did not have any psychologic problem, but some kids did and left school finally.
She was accepted by four ivy league schools and will graduate from Harvard this coming summer.
If your daughter does not care peer presures and can be own whatever she wants, a good private school would give her more benifits than public school. She can be a number one easily there according to your descriptions above. Good luck.