And the fact remains, the public schools topping the list you linked are selective schools - schools that select children based on excellence and giftedness. By definition they select the top performing and most capable students. Given private schools don't just 'select' students on academic ability they are unlikely to out score selective schools academically.
Private schools may very well offer better academic outcomes for regular students than a local public school and that may be why some parents choose to send their children. But I'd never expect them to compete with academically selective schools and am surprised anyone would.
I'd hazard a guess though that people spending upwards of $20k per child on private education isn't just after academic achievement for their child. They want the facilities, the community, the experiences and the connections.
As a non-native Sydney person I'm constantly amazed at the emphasis Sydney people place on what school someone went to. A person can have multiple degrees, have been a Rhodes Scholar and had twenty years of high level corporate experience and STILL get asked what school they went to. It's remarkable.