going to a slightly better state school does not somehow magically improve your chances of a top job compared to people from other schools who are as good or better than you
What do you think that "magic" is? Seriously, have you thought through what the mechanism would be? 
Personally, I think there are several separate reasons why the outcomes between state and private are skewed: each reason contributing a part of the difference:
1.Family networks directly giving access to certain jobs aka old boys network. Would still happen regardless of school, and only applies to the very wealthiest/most influential families. Certainly no-one I know at DD's school.
2.Candidates having the 'social language' (common frame of reference and life experience, same accents etc) that makes them 'comfortable' and so seem a safer bet to employers from a similar background Thankfully less common these days, with focus on diversity. Also note that it works in both directions, eg trade apprenticeships avoiding 'posh' kids. Where it still exists, it's mainly from family, rather than school
3.Family wealth giving access to unpaid internships which lead to those jobs Nothing to do with the school, just family wealth.
4.Parents are usually stronger academically and wrt job competence than average: in order to hold the type of jobs which pay enough for private school. This results in a different ability distribution between students at state and private school. Of course not all private school kids are more capable that all state school kids. Of course there are plenty of bright, capable parents in low-paying jobs (possibly very important and fulfilling jobs), or who earn enough for private but choose state, or kids in state much more capable than their parents were. Of course not all private school parents are capable of professional jobs (may hold them through family privilege - but that's a small number), or some private school kids are paid for out of family money, or kids in private much less capable than their parents were. Of course I'm not saying that all private educated kids are more capable than all state educated kids - that would be ridiculous, and is in no way reflected in outcomes. But it's pretty obvious that if you remove most of the left hand side of the ability curve from private schools, then you have to expect that a higher proportion of private school kids would be more successful even if everything was completely fair and equal. So the distribution is skewed, but the school makes no difference to any individual child's chances, since it's basically their family genetics and upbringing.
5.Good education does actually help kids to improve as people and better reach their potential. When it comes to this part of the difference in outcomes, it isn't that private school kids are taking opportunities from state school kids who are better than them - it's that they have actually become more capable than if they had gone to state (whether academically, in social ability/confidence, or in perspective/attititude eg willingness to take risks, or in their expectations of what they can aim for). So they get opportunities - and make opportunities - ahead of state students who would have been the same as them had they gone to the same school. I'd argue that this isn't an unfair advantage. Certainly no more than the advantage you give your kids by reading to them, modelling good behaviour to them, giving them opportunities to discover the world like extracurricular activities and travel, feeding them well and giving them a stable home. In fact, all of those contribute much more to their life chances than education. But all of these things are a social good, making better, more capable citizens.
This last reason - good education actually improving the individual - is the only one where the school makes any difference! (All the rest are correlation, not causation) And it's why I've got my DD at private school.
tldr; As a pp said, don't confuse correlation with causation.