I'm the product of the Australian private school system and I'm not sure it's relevant to the UK situation. As someone said, we have Catholic schools which are very cheap (just hundreds per term, in some cases).
The Australian university experience is also very different whic I think is a much bigger influence than private vs public. We have a vocational system - you take a subject at uni that is directly correlated with your intended career plan. We don't have entrance interviews but instead receive an overall entrance ranking that means a computer determines whether you are eligible to enter a certain course at university.
People who miss out on a popular subject can trade up to better courses once in university e.g you miss out on physiotherapy so enrol in a general science degree and assuming you pass, you'll have enough points to start physiotherapy the following year, as moderately good passes at university are equivalent to top Year 12 results. This also applies to transferring between universities, so you can trade up to a sandstone university if wanted. This massively levels the playing field at university and therefore the graduate jobs market.
I actually don't know that many people who finished the course they started at uni; the majority of my friends traded up.
The UK seems to set people up for failure from 15 - poor GCSEs limits your sixth form college options; poor A-levels limits your university course choice; weak university degrees mean that a whole swathe of graduate career options aren't realistic. We seem to have a lot of intermediate steps to redeem ourselves in Australia.