Despite sending both my children to private secondary (for a variety of reasons, SEN provision being one, having been horribly failed in the state sector), I don't think it's going to automatically propel them into a successful life. When it comes to work experience and contacts, I actually think we as parents (and the careers of our friends) will give them an enormous leg up - contacts in Big 4 accountancy firms, Bank of England, top tier law firms, ad agencies, TV production companies, high end retail, aviation industry, doctors, teachers etc etc etc.
I fully acknowledge that they will be lucky to have this leg up, but it will have little to do with their schooling, everything to do with their social 'standing' (for want of a better word). They'll have grown up with their 'normal' seeing the adults around them working in household name companies - that's an inequality very difficult to rectify. I know there's some great work where companies are trying to do mentoring schemes and similar in more disadvantaged areas, but I think there's so much wrapped up in a person's norms, it's such an enormous hurdle.
I'm not saying it's right, but I'm suggesting a private education in and of itself isn't really what's causing inequality in society. We talk about white privilege when discussing race, and this can be extrapolated so much further. Yes I'm trying to do the best by my children by sending them private (and thankful we have choices available to us), but I'm also passing down my privilege of a good upbringing/education/career, which was passed down from my parents' achievements etc.