What you have to think about is private schools are a business, the schools cherry pick the best students and go to great lengths to teach and coach students to keep up the their intake into university. The public schools get what is left with many from underprivileged back grounds yet according to the NAPLAN tests shows there is very little difference between a public and private school.
The best thing about private schooling is the 'old school tie' my eldest attended Haileybury College, still as thick as two short planks but landed a plum job because of his school.
What happens when they get to university?
State school kids do better at uni
As for leadership can it be taught, only to a certain level, a lot of the traits of a good leader one has to be born with. We often see people in politics who can talk the talk but fail to walk the walk. It takes that bit extra to be thick skinned and tough, while it can be taught many just can't carry it out.