THERE is no doubting that private schools are a breeding ground for privilege. Anyone lucky enough to afford the fees can rest assured that their child will be given a lengthy headstart in life, with a place at a top university then a well-paying job theirs for the taking. You just have to look at the CVs of our MPs and judges, CEOs and – yes – journalists, to see how over-represented those with a private-school background are.

But while this is clearly unfair – while it perpetuates elitism and allows social exclusion to thrive – is abolishing private schools really the best way to, as Labour’s Shadow Chancellor John McDonnell said at the weekend, build “a more cohesive and equal society”? His party certainly seems to think so.

Responding to a motion put forward by Unison organiser Ryan Quick, delegates at the party’s annual conference voted on Sunday to integrate private schools into the public sector as a means of loosening the grip their old boys have over society and making the “wonderful resources” they have at their disposal available to all. To achieve this, Labour would remove independent schools’ charitable status – something the Scottish Government has been consulting on for much of this year – redistribute their endowments, investments and properties to the state sector, and require all universities to take just seven per cent of their students from independent establishments, reflecting the make-up of the wider population.

It was just the kind of posturing that Corbynistas have been crying out for and, with the anticipation being that the policy, if ever enacted, would spell the end of private schools altogether, the Abolish Eton campaign greeted it with jubilation. Campaign organiser Holly Rigby, a state school teacher who had helped push the issue up the party’s agenda, called it a “really big victory for the left”, adding that “this is a really positive moment for the Labour Party because it’s the strongest commitment that the party has made to deal with the problem of private schools in a very long time. It’s a really radical commitment as well.”

But how much of a victory can Labour’s motion really be when all it does is make private schools the problem that needs to solved rather than addressing the very real issues that hold back too many in the state sector too? Is the policy really worth crowing about when, far from making all state schools that little bit better, it may actually stand more of a chance of making them that little bit more crowded, that little bit more stretched and so that little bit worse?

It is a lie perpetually spun by those running private schools that what they offer is a superior education when in fact all they provide is an expensive, resource-rich and selective one. Indeed, study after study has shown that when things like socio-economic status are controlled for, the type of school a child attends has little or no impact on their academic attainment. Yet while it would be nice to imagine a wider distribution of the little things that set private-school kids apart – the polish of one-to-one tuition, the confidence of knowing your opportunities are endless – the opposite is likely to be true in a state-school-only world. Parents, after all, are hardly likely to continue paying for an education that has been taken over by the state, while without parental funding the ability for schools to provide those things will disappear.

Given that state-school kids are every bit as capable of achieving academic success as their privately educated peers, some may feel that removing the added extras that money has bought for the latter may be enough to level the playing field in and of itself. Nothing could be further from the truth, though. Because while capability is one thing, actuality is another, and the vast differences that exist within the state-school sector are way more pernicious, and way more worthy of addressing, than those that have grown up between it and the private one.

Indeed, while it cannot be right that the universities of Oxford, Cambridge and St Andrews – all of which are renowned for propelling graduates into the best of careers – take 40per cent of their students from private schools, neither can be it right that state-educated children in the poorest parts of the country are left trailing behind those from wealthier backgrounds. The fact the Scottish Government-funded Growing up in Scotland study found children from low-income families to be 13 months behind their more affluent peers in vocabulary skills and 10 months behind in problem-solving skills on day one of school is unacceptable, but closing down private schools will have no positive impact on that.

Yes, Labour’s plan to require universities to address their admissions policies is long overdue – and some institutions are already taking action off their own bats to do just that. But unless something drastic is done to close the poverty-related attainment gap, only wealthier state kids, whose parents may well have paid a premium to live close to supposedly "good" publicly funded schools, will stand to benefit. And while seizing and selling swathes of valuable land owned by the likes of Eton College may seem like a good way to fund the closing of that gap – and it would certainly win the votes of those on the hard left – does anyone really believe they would give it up without a long and costly fight? What would happen to attainment in the meantime?

There is no doubt that British society is as unequal today as it has been at any time in its history, with the rich continuing to get richer while the poor go on getting poorer. Private schools, which send their pupils down gilded paths that equally bright and equally talented children from poorer backgrounds are too often excluded from, undoubtedly have a part to play in that.

But the solution to social inequality does not lie in eradicating every privilege enjoyed by the wealthy when they will simply be able to use their wealth to find ways to overcome that. Yes, it is right to expect private schools to pay their fair share of taxes and, yes, universities and workplaces should be required to reflect the make-up of the population at large, but abolishing private schools altogether? That’s just a distraction.

If the Labour Party really wanted to be radical, if it really wanted to score a big victory for the left, it would focus on finding ways to make schools at the bottom of the educational tree better, rather than trying to make those at the top worse.