Nanban, Switzerland is hardly known as 'communistic' and yet this is exactly what happens - no choice, and it works really well. That way you don't get 'excellent schools' and 'sink schools' and children learn to grow up in a mixed environment, rich, poor and in-between- which stops stereotypes being formed in childrens' minds and finger pointing. When you grow up in a mixed school, you know right from the start that stereotypes don't apply in 90% of cases - and it really helps society as a whole.
As said, the UK has had such a long tradition of (so-called) 'public', private and religious schools - and I just do not know how this could be addressed now.
Totally agree that the shenanigans and border fraudulent behaviour of many to get their kids into the best of state schools- is in many ways 'worse' than paying for private education. See the example give re my niece on a previous post. I have of course not discussed this with my sil and bil, nor my niece- as it would cause upset, but I just couldn't believe they would take such steps.