"In areas like Richmond isn't the private number of children educated supposed to be nearing 20%?"
I'm not sure of the exact figure, but that sounds about right.
However, AFAIK, currently Richmond and surrounding boroughs are struggling to provide state places for the children that they do educate (several years of temporary classrooms in Wandsworth, Kingston, Richmond, no funding yet available for secondary school places for current 'bulge' years), and this is a problem in other parts of the country too.
The problem with educating children, is that whatever your principles, you can't educate your child in some far off future when the government might provide the education your child needs, you have to educate them now.
As far as I can see, the only way that it would be possible for private schools to be abolished would be huge increases in tax for everybody, including those who currently benefit from a good local state school. I'm sure many parents who feel forced into sending their children to private schools would be very grateful for this more equitable funding of their child's schooling.