Remember Nick Clegg? Back in 2010 the then deputy prime minister declared that reducing inequality was for “old progressives”. For “new progressives” – such as him and his progressive mate, David Cameron, the priority was “reducing the barriers to social mobility”, primarily through education. Now Justine Greening has become the latest in long procession of ministers determined to improve social mobility. In her case it’s by making it the “guiding mission” of the Department for Education to help the talented to the top. What this means is: equality is dead; long live sharp elbows.

But now finally there’s a challenge to all this. Labour’s 2017 manifesto notably omits any mention of social mobility from its educational goals. Instead, it emphasises fairness. Jeremy Corbyn’s team has grasped two facts that elude most politicians. First, education alone can’t effect social transformation. The Social Mobility Commission recently revealed that two decades of educational strategies to improve social mobility have failed. Those who assume education can bring about social mobility hark back to what Angela Rayner, the shadow education secretary, has rightly called the “mythical golden age” of postwar grammar schools. It is true that between 1945 and the 1970s a larger proportion of children ended up in higher social classes than their parents than ever before or since. But this wasn’t thanks to grammar schools, whose very few working-class pupils were disproportionately likely to leave by 16 with minimal qualifications. What made the difference was Labour’s investment in public sector employment. Educators can’t create jobs; governments can.

Labour also understands that social mobility is an undesirable educational goal. In the postwar years, opportunities in the professions and other well-paid, secure jobs expanded, benefiting huge numbers of people. But today, social mobility means a scramble for the few jobs that offer security. Educators are expected to identify and help those intelligent enough to merit a top university place and top job. But there’s no robust measure of intelligence, which is now widely accepted to be situation-specific and to develop throughout life, not be fixed at age 10 or 16. The majority of children on “gifted and talented” programmes introduced by New Labour are from middle-class backgrounds. In an unequal society, those with something to lose do everything to maintain advantage for their children.

Most seriously, social mobility reinforces social inequality. Policymakers inaccurately equate the two, but the social mobility agenda assumes we’re stuck with a hierarchical society. Its supporters uncritically accept that there are “top” universities – the Russell Group – and “leading professions”, defined by Greening as law, medicine and banking (notably, education, meant to deliver so much, isn’t a sector that the talented are encouraged to enter).

This approach has served most educational institutions badly. The focus on widening access to Russell Group institutions has had no discernible effect on their student bodies but has ensured politicians and the media have overlooked the gross underfunding of adult and further education. And the social mobility agenda ultimately reinforces the Russell Group’s claim to represent our “best” universities. This has been unquestioningly accepted by politicians, perhaps because so many of them were educated at these institutions. In reality, the Russell Group is a self-selected band of university managers named after nothing more highbrow than the hotel in which they first met.

Of course there’s much more to be done to ensure working-class children are able to enter our most socially exclusive universities. Labour’s plan to abolish tuition fees and reintroduce maintenance grants is an excellent start, and has already transformed the political debate over when fees will be raised into a discussion of how long they can survive.

But Labour’s national education service proposal goes further still. The social mobility agenda has been lamentably unambitious. Its focus on the talented few offers no hope for the many. Its narrow focus on employability compares badly with Labour’s emphasis on lifelong learning for skills, creativity and cultural enrichment. By asserting that fairness and comprehensive provision are vital educational aims, Labour is offering a radical alternative. The Labour frontbench, several of whom are alumni of adult education, FE colleges and polytechnics, aren’t content to simply focus on getting a few children into the supposedly top institutions. Instead, they are inviting a national debate about what constitutes a good education, and how all of us – young and old – can enjoy it.