"You have to pay more... therefore we can assume private schools provide a better education".

1) a mathematical answer: remove funding to the private sector, you create a crisis in the public sector as many thousands of students would flood in to the public system requiring a massive taxation increase and infrastructure expansion. Subsidising the private sector frees up funds for investment in public education; 2) a philosophical point: diversity in education enriches society culturally and allows for innovation and experimentation in education; 3) an obvious point: most private schools do not cater to the rich, they cater to community values like religion etc which not everybody wants for their own children; 4) A seldom acknowledged point: Expensive private schools typically have significant numbers of non-fee paying students from disadvantaged backgrounds and serve a precious role in educating country significant numbers of rural Australians from districts where education at a local level faces significant hurdles.

Good question. If private schooling is a lifestyle choice, fine. But it should not be a quality issue. Let's see an end to subsidies for lifestyle choices, silly decisions like dumping computers in classrooms without a plan for their use, etc. and put the focus back where it belongs: well qualified teachers and smaller classes.