It's not about 3* better, it's about having the resources to devote attention to each child. My children are privately educated with 15 in a class and on top of that they get one-to-one attention to their weaker areas.
My son when he started at private was a long way behind, we were told by the state school that he was doing fine no concerns at all, but the private school said 'he's miles behind, we might have to keep him behind a year'. In the end he was fine, and now he is on track for really high-grade academic excellence. Now it's of course possible that he would have eventually made it at state school, but for the time he was there, at a 'Good' state school, he wasn't even on their radar.
I was in the top set at a posh state comp, we had 30 in the class and I found it quite unstimulating. I've been round some top rank private schools and there is no comparison whatsoever, and the opportunities for bright children to be stretched is simply incomparable.
No you don't just turn money into results, you use money to buy more teachers, more clubs, more books, more extension materials, more learning support for those who perhaps wouldn't be on the radar in a less-well-resourced school.
And it certainly is substantive knowledge, my DS is 11 and the substantive knowledge in many subject areas is far more, er, substantial, than what I had at his age.
It's ignorant bollocks to suggest that companies want to recruit 'one of their own'. Oxbridge degrees will help, certainly. Many companies go to recruit directly from Oxbridge, because if you're running a biotech firm, or a hedge fund, then you want to employ geniuses, because they will make you money, and you are far more likely to find these people at Oxford than Oxford Brookes, say.
What the parents' money and connections can certainly do is help find 'Tim-nice-but-dim' an undemanding job somewhere, if all else fails, but that's not what you are setting out do by buying a private education.
Rather, what you are looking to do is to ensure that your child gets good exam results, and is a good employment prospect (there are plenty of young people who don't have basic skills such as presenting themselves properly, turning up on time, having a conversation with the interviewer). In addition, concerns such as funding for university and any work training also evaporate.
These are huge advantages, but the goal is still education rather than nepotism/cronyism.