The issue is actually "unfairness": That private schools give an unfair advantage to their pupils over state schools. Is this actually true - and what sort of advantage is it? If it is academic then why can't we use the same methods to bring state schools to the same level? If it is just money, why can't appropriate funding be found for state schools? Or maybe it is something else? If so how can that be addressed?
There are a number of different elements, some of which are present in some private schools and others in others. In purely academic terms, there are a number of studies which show that some private schools do manage to establish a better academic performance, on average, than their state sector equivalents, primarily through maintaining very small class sizes, but even that is limited to more capable pupils - that method isn't particularly effective at improving the performance of the less academically gifted. In general, it seems, there isn't a huge amount of increased academic performance from private education - some private schools are selective, which has a tendency to improve their league-table performance on results achieved, but they don't score particularly well if 'value-added' is considered (although there are a number of debates about how effective we are at measuring that).
Typically private schools offer a broader curriculum - certainly things like arts, music and sport are typically better funded than state schools, perhaps part of the reason so many Olympians and actors in recent years have emerged from the private school sector.
Both of these - smaller classes and a broader curriculum - tend to cost more per pupil to maintain, so funding is at least likely to be part of the issue. That said, there is also a flexibility private schools have with respect to if and how they teach against the National Curriculum - academies and free schools have a similar freedom, as I understand it, but typically choose not to exercise it particularly heavily except in the area of the faith schools who diverge on RE.
Within the teaching profession - from my contacts with teachers - there are some teachers who would never go near the private sector, but there are significantly more that would, and the private schools tend therefore to be able to pick and choose their staff more selectively. They don't necessarily offer better pay, but they offer a better employment environment and better working conditions and (tellingly, according to some teachers) freedom from Ofsted. Having the freedom to pick better teachers perhaps helps.
As a parent, what particularly appeals to me about private schooling is that you tend not to have disruptive, aggressive, bullying children - whether it's because the parents who are interested in paying for education are likely to have brought their children up better, or the teachers are able to more effectively manage the pupils I don't know, but the exclusion rates are typically lower than state sector and yet the issues don't seem to arise.
Overall, I'd say, there's a slight academic advantage for the particularly capable, but there's a better atmosphere and broader education that seems to stand privately education children in good stead. That doesn't, in any way, justify the sheer rate at which they are over-represented in places like Parliament, the big four accountancy firms, the Judiciary, Oxbridge and the like; in those instances, though, although 'private schools' are accused it is in reality a very small subset of those schools that are overrepresented. Rather than looking at private schooling, I think we should be looking at the selection processes for those places to see why things are skewed as much as they are.
O.