@stuckdownahole - I haven't had time to read the whole thread but I think I agree with you, but only up to a point. I think that our own personal decisions are material to our political beliefs and vice versa. I think this is about principles and we have reached a strange place in our politics and society where individual principles and morals are somehow seen as irrelevant to public office and detached. I think that is massively problematic for all sorts of reasons. See though: Boris Johnson.
The trouble with private education is that people will always find a way to justify something for their own child which is fundamentally unjust and unfair, and which damages the prospects of others. A commitment to private education is in fact a dereliction of the common good. Someone else said we should make state schools as 'good' as private schools but even if that were possible it is missing the fundamental point - education and all the cultural resources that come with it are not that valuable in absolute terms, they only matter in relative terms - in simple terms, it doesn't really matter what you know, just that you know more of it than the people you are competing against. But this isn't just about knowledge. You could level-up state and private schools (though that's not happening and is unlikely to do so) but people at the top of society would find other ways to keep these rewards to themselves, by changing the rules (around what is seen as valuable and why). As just one example, when more people started going to uni, the professions did become more open, so to manage this, it suddenly became important to demonstrate all sorts of other sporting accomplishments and leisure interests, which are often expensive and largely available to the middle-class (and especially at private schools). There is nothing about these interests that make people better at professional jobs, they are completely arbitrary. But making them important (changing the rules) restricted entry once more to more privileged people in ways which seemed fair (but were not).
Other posters who have wondered about how social mobility works have to remember that a political commitment to social mobility is not about making people the same - it is about legitimating extreme hierarchies of income, status and wealth. In other words, if you have a great deal of money you can make this seem acceptable by claiming that similar rewards are available to anybody on the basis of talent, hard work and skill. That is the basis of the merit principle which has never been realised in the UK - in fact we are miles away from it. So it works as a useful myth.
Basically, the very foundations of social mobility are political and they are located on the right. On average, people whose politics are towards the left are much more interested in inequality which is about reducing hierarchies and making sure that everybody can enjoy dignity and respect.
Inequality and social mobility are not oppositional though - social mobility is more likely in more equal societies, but the current government appears to have lost sight of that fact. 