The royals and their children ect, are ambassadors for the country, why should they not have the best education if the options are available.
Yes in an ideal society everyone should have an equal opportunities education.
But if that's your starting point, what do you do with child a, that is better at maths, child b, is better at psychological perspectives, how long before people then say hold on child, x, y ,z is doing better than child a, ect.
These days if the determination and drive for learning is there people will learn, and no matter how much level field the schools could begin with, say their is only state schools, for all, you would then have different outcomes still as some parents would be more invested, some would hire tutors ect, it could be debated that schooling can and will never be truly equal for all.