Not all private schools function like Haileybury though; it's true that there are a fair few kids at the top end who get "bought" in by scholarships, but examples like Haileybury's buying of whole teams are very rare (for reference Haileybury is often considered a joke in regards to its scholarship teams for sports). Generally speaking, for instance, of the 10 or so kids offered entrance scholarships for academics, only about 7 of them would be in the 25 students scoring 99+ ENTERs each year at MGS.
The other thing is that in my own observations, a lot of the scholarship kids probably would have gone to private schools anyway. There are a few kids who most likely wouldn't have been able to afford a private education without their scholarships, but at the same time about half of the scholarship kids in my year went to private primary schools as well.
That aside, I've spoken to kids who have moved from public schools to private schools, and in their words the private schools are a lot more accomodating for "excellence" in regards to atmosphere. I agree with your statement about the students making the school; the thing with private education though is that you are actively encouraged to make the most out of your schooling, and the fact that your parents are paying about $20,000 a year really does weigh in on students' minds, hence the high ENTER averages, etc. Not only that, but the kids who moved between schools said that private schools kids themselves will be more supportive of their peers in regards to achieving - one student complained that at her old school, "smart" people were made fun of, whereas after she moved she found that there was a lot more encouragement for her to do well from peers. Whilst I can't say for certain, I'd wager this is similar to selective public schools, perhaps even moreso because it's often the kids who try to get in as opposed to the parents forcing them to.
Perhaps parents will often choose private schools because of the prestige attached, but the reality of private schools really do justify the costs - I came from a public school background as well, and I personally see very clearly the effects the complete change in environment has had on me as a person.