That's bonkers. The state education system is for the children of the state. We could possibly avoid private if we went without holidays, stopped all out-of-school activities etc, but I wouldn't want to. Private schools are an option open to some parts of society who can afford them. But they are luxuries, not really any different from things like ski-ing holidays, private health care, expensive cars, designer clothes and so on. I wouldn't feel I had to go on a holiday or buy a bigger car just because I could afford one, and I wouldn't want to force my children into a more narrow experience of society than the perfectly good one for which my taxes currently pay!
If you did impose this there would be a lot of movement in an out of private and state schools. I know plenty of people whom have chosen state schooling. Their local state school is excellent. The catchment area is small and the housing is extremely expensive - £600-£700 for a semi and we aren't in London. The school offers no wraparound care at all as the pupils are collected by their parents (mostly sahm).
Ds started his education at a private school for many reasons the main one of which was nothing to do with the education he would actually receive. If I could have found the same (non-education) provision in our local state school then that is where he would have gone.