I went to a good state school (grammar, and academically good). I did very well and went to Cambridge. I met my DH there, he went predominantly to independent school - he moved around a lot so went to several different ones and also did half a term in a local comprehensive. I have taught in state and independent, he teaches in independent. So we've got quite a breadth of experience between us.
As many have said, there is a LOT of variety between schools of both kinds. A child that is supported by parents and encouraged to have confidence in their own ability to learn and succeed will do well academically in either environment. However, smaller class sizes, (generally) less stressed teachers and a larger dedicated pastoral team means problems are usually picked up much more quickly. This means they don't go unnoticed and cause academic problems, but I have seen instances where I think the child would have learnt more by working it out themselves - this has led to some less resilient kids feeling that they cannot solve their own problems, which has given them some trouble at university.
The extracurricular opportunities at independent school at astronomically greater. My DH has experience of just about every sport and is reasonably good at them, and can do lots of other things he tried at school. I think this is one of the things that helps with networking - having had a variety of different experiences means you are more likely to be able to find common ground with someone in conversation.
As far as experiencing a wide variety of different people - financially there is a huge range in independent schools IF it has a decent scholarship/bursary program (which many do, especially in London). There are usually Lso many students from other countries, which provides some breadth of perspective. I have found that the super rich pupils you get at some independent schools aren't at all snobby - they have never been shot of money, so almost don't value it. There are a few more of the more obnoxiously entitled kids than I saw at state school, but actually not all that many, and that attitude of entitlement is something kids need to learn to deal with as they will definitely come across it at university!
Overall, I'd say it's a benefit academically, especially to those who are less confident or able at age 11. Emotionally I think it provides MANY more opportunities for speaking and expressing, which builds confidence. BUT parents need to be very careful that children don't feel pressured unduly and are given opportunities to fix their own friendship issues and conflicts.
I would very much like my DD (& any other DC we have) to attend an independent school, but would select it carefully and try to speak to existing parents first.