I was giving this some thought and basically why not?
I think security is a red herring because fundamentally security arrangements must be as easily arranged in a state school as a private school. Additionally privacy is a tenuous argument given that the children are being taught in classes regardless of schooling rype.
I think there is a real opportunity to modernise the royal family at least on this point as there is no specific constitutional arrangement that means the royal family have to be home educated or educated in the private sector; in fact this may be a tradition more honoured in the breach than by its observance. I think there would be an increase in popularity of the family of they are seen to be educated in a similar fashion to the cast majority of the population and therefore gaining a greater connection to the public. It is not as if schools like Eton have a divine right to educate royalty and in fact the royal family have went to a diverse range of elite private schools.
There are some anomalies with this education tradition as when a member of the total family ain't particularly academic then it becomes a challenger where the majority of their peers are (Prime Harry). Of course university in this country is non private but the system of educating the royal family possibly means there are only certain 'acceptable ' universities for the royals..
we don't want to leave the impression that it is only acceptable that the royal family mix with a certain social cohort at the formative schooling period and this doesn't have to be the.case. Did King Charles gain from a private education? From what I have read he was rather miserable at times so private schooling doesn't naturally lead to happiness as well as the fact l that educational qualifications do not influence the royals' future as much as for the common peiople.