The quality of private education - as with all education - here in the UK really does depend on the area one lives in IMO. I live in south east/greater London and most of the private schools here do provide a better quality of education than the state/public schools, however I went to a private secondary school but left halfway through to attend a state school because the education there was very poor - the state school I went to was a lot better and I gained higher exam results there than I did at the private school I'd previously attended (and my state school was "only" rated Ofsted good). At the private school I had attended, teachers were always leaving and there was a clear student bias in the teachers and pastoral care was pretty much non-existent. I've completely lost my faith in the private system as a result, and will be sending my future children, if I have any, to state schools ideally.
But, that won't be the case for any royal children as they will be sent to some of the country's best schools, where they're guaranteed to get a much better secondary education than I did!
"For beautiful eyes, look for the good in others; for beautiful lips, speak only words of kindness; and for poise, walk with the knowledge that you are never alone". Audrey Hepburn
"Think of all the beauty still left around you and be happy". Anne Frank