Firstly in many areas it isn't a tiny proportion. In a couple London boroughs it is between 40 and 50%, and in many others it is still very high. In these areas private schools completely split society in two, where the rich and the poor kids barely mix, at all. If all those children attended the local schools, the local schools would be just fine! A normal mix of London society. Instead in some areas there are sink schools and private schools, and those who can't go private who can afford to move, move! How is that good for social cohesion or creating a strong community that supports the local people and local schools?
When I referred to some children not experiencing society I actually meant the private school kids don't get to see real life, not the other way round.
Also I didn't just refer to private schools but also grammar schools and changing admission criterias to not be based on distance for state schools, to avoid this rugby scrum of scrambling to buy overpriced housing on the doorstep of a state school. Those three things combined could have a huge impact on evening out the quality of education across the board.
Your point that most private schools aren't like Eton is totally beside the point. The fact is that most of the people RUNNING our country have no experience of what it's like to live a normal life in this country. Never needed state education, never needed the NHS, never needed benefits, the list goes on, some of them are just thoroughly unpleasant people. Yet purely by attending prestigious schools that most children have no hope of attending, they get a leg up, they get all the top jobs. Wouldn't we rather see more balanced people running the country? People who got there on merit?
And the comment about 'oh but some will go abroad...some will homeschool...' fine! They are free to do that. But most children want to go to school and benefit from being in school, and most don't want to be sent abroad away from thier family and friends. If schools were all largely fine, they wouldn't need to. And over time it will just become to the norm.
It's not imagining a 'utopia' to want to see positive change. It's not something that can be fixed overnight, but nothing ever can, it's about making positive gradual steps in the right direction.