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The main thing that state education fails to offer to students is the 'networking opportunities' prevalent in the private sector. The parents of the students in these schools have rarely 'raised themselves up', but are merely the perpetuated product of these schools themselves. This is what stifles opportunity, a system that maintains the status quo through nepotism. Occasionally people break through, but these are still statistically in the minority.
As for the effectiveness of state education it costs around £6k per year to educate a child, with average classes of 20, compared to £12k per student in the private sector with average classes of 10. Educationalists agree small class sizes are key to discipline, progress and achievement. So if we double the state funding then we could actually compare the two, which we can't at the moment. Increase this to £30k a year (The Eton price tag) and we could get it down to a handful of kids in each class, then we could have a real comparison.
Smaller class sizes would make a real difference, something that won't happen with public spending cuts. This being said small classes don't benefit everyone, just look at Prince Harry. That being said, the state education system could just cheat for students and use the extra money to fund law suits against art teachers.