Eaton would not exist in its current form if private education was abolished. You could have streaming or even special state schools in which the especially gifted received a faster moving, more advanced education, but, if you do that, then social background should be taken into account when assessing who the especially gifted are.
So you think we ought to have state-sponsored 'exclusive' schools? How would you decide which child was most suited to such a school, by the way?
By the way, the 'best' schools aren't those that necessarily achieve the most GCSEs or A-levels. Rather, they are those schools that raise the educational achievement of the children they have most. As a result, most of the truly 'best' schools are in some of the most run-down and deprived areas of the country.