For some reason the right like to see equality as being Equality of outcome, it is not it is Equality of Opportunity.
Everybody regardless should have the same equality of opportunity and then you may actually have a meritocracy rather than the sham meritocracy that we see under inequality of opportunity. There is nothing meritous in getting a top job if you have a head start in life, there is a merit in getting a top job if you have do not have a head start in life. The problem is too few who dont have the head start ever have the chance to become meritous. The top jobs are reserved for an elite, an old school tie network who have got the job by deem of inequality in opportunity.
If we want everyone to succeed then equality of opportunity is a must, it then leads to the each according to his ability , each according to his needs and society based on fairness and equity just like my fellow comrade and ex-roty Metalblue desires.
I have no problem with private schooling, i would make all schooling better than private schooling and it would die a natural death as it only exists to further inequality of opportunity, once that inequality is erased, then there is no need for the schools, because everyone gets an education suitable to their ability.
That then will be good for democracy, as we would no longer have a narrow cabal of people running the country and we might actually have some really bright kid from a Salford council estate as PM rather than an Eton muppet and his gang of cartoon caricatures, cactus plants and frozen paving slabs.