You can't say it's only attending private schools which gives wealthier kids an advantage. They'll also have bigger houses, better holidays, private lessons in any activity they want, the latest clothes and accessories - not to mention better healthcare, if they go private.
Even if they didn't go to private schools, kids from well-off families would still have an advantage in terms of education. They'd have had more opportunities to develop their skills away from the classroom: and, if they've been surrounded by wealthy people all their lives, they'll often have had more encouragement to work hard in school and get a high-paying career themselves.
I think it's a lot more difficult for kids from rough areas to believe in themselves and aspire to getting a really high salary - if you came from a family like my dad's where no one stayed in school after 16 and who were largely long-term unemployed, you'd have to be incredibly self-motivated and hard working to stay in school and try and make a better life for yourself.
Overall: just being well-off gives kids advantages that are "unfair". I'm not saying it's morally right that some people are given more chances than others, but that's how it is and that's how it's always been. Though I think grammar schools - like the one I went to - are the best option, while the school system remains as it is I can't blame any parent for sending their child to a private school. Any good parent wants to give their kid the best start they can, and, when I've got kids, I'll probably send them to private school if I have the money.