It doesnt work. Look at the FTSE 100 CEOs 26% were privately educated, thats against a figure of only 7% attending public school. Its even worse when you look at the the media which has some of the highest numbers of privately educated people. Of the 100 most influential news editors and broadcasters, 43% went to fee-paying schools. Similarly, 44% of newspaper columnists were privately educated, with a third - 33% - attending both an independent school and Oxbridge.
Whilst some of the power is in publicly funded professions, the real power is in banking and private multinationals. They influence policy decisions and the media decide what narrative to play.
If I look at the company I work for, its a multinational business and the higher you go up the tree, the more privately educated people there are, to the extent that at board level there is only one person who didn't go to a public school.
Whilst state schools are clearly underfunded, one of the biggest costs is providing support for kids being brought up in poverty and the problems that it brings. Reducing poverty is the solution to better schools as it is for most of the issues with society. More could be done by increasing taxation on wealth and/or income, but of course nobody wants to talk about that as it wont win votes. Surely on that basis taxing public schools is the token gesture or sticking plaster and an ineffective one at that.