“For old progressives, reducing snapshot income inequality is the ultimate goal. For new progressives, reducing the barriers to mobility is.” So said deputy prime minister Nick Clegg in 2010. In the decade since, social mobility has been a favourite theme for those on the liberal wing of the government – both in the coalition and the Conservative administrations that followed it. Policies such as school academisation and increasing home ownership were promoted as assisting children from poorer backgrounds to get ahead and breaking the stranglehold of the middle and upper classes on the top universities and jobs. But that doesn’t mean these initiatives were right. When the government’s own experts say inequality is now “entrenched from birth to work”, it could not be clearer that its approach has failed.

A new joint report from the Sutton Trust and Social Mobility Commission is the latest proof that far from becoming a more socially mobile society, Britain remains stuck. The study of 5,000 members of the elite (judges, politicians, top business people and pop stars) shows that private school pupils and Oxbridge graduates continue to monopolise senior positions. The findings come as no surprise, given a plethora of recent evidence pointing to the same conclusions. The previous set of commissioners, led by Alan Milburn, walked out in protest at the lack of progress in 2017.

The report makes useful recommendations. Contextualised school and university admissions, and a new focus on segregation in education, could help break down divisions among children and young people. Such practices should continue in the world of work. Employers should monitor socio-economic backgrounds in the same way that they monitor differences of sex and ethnicity.

Yet while the principle that people should be mobile – in the sense that they can acquire status they were not born with – is sound, the emphasis must shift. This month, Labour promised to abandon social mobility as a policy goal and replace it with social justice. This new and more egalitarian direction is welcome. Equal opportunities and access are important, but insufficient. Justice is more than social mobility with teeth. Policies that attempt to bridge and repair the divisions and rifts that deface British society are desperately needed. Growing inequalities, particularly of wealth and geography, are among the causes of the political crisis brought about by the vote for Brexit – and are now being exacerbated by it. Boris Johnson’s promise of tax cuts for the rich can only make a bad situation worse, even if he has now modified this with a contradictory commitment to prioritise the poor.

The intergenerational lens is a relatively new but essential tool in the policymaker’s toolkit. That’s because millennials (born 1981-2000) have not seen the improvements in living standards – compared with their parents – that previous cohorts took for granted. Home ownership rates in particular have fallen sharply, as has the proportion of younger adults renting from social landlords. The result is many more families trapped in high-cost private rentals, with all the insecurity (including for school-age children) this entails. As inheritances are passed on, including by the 11% of adults who own a share of a second home, the gaps will continue to widen – and an already widespread sense of social injustice will continue to mount.

How a rebalancing is achieved is a key question not just for the next prime minister, but also for the country. The social care crisis means new taxes on wealth are inevitable sooner or later. Far less certain but just as necessary is the kind of political leadership that will enable us to come together again as a society and rediscover some of the common bonds that we are in danger of losing.