With the threat of a hard Tory Brexit and crumbling public services, to be distracted by Labour’s internal divisions this election is to focus squarely on the wrong thing. But there’s one dispute that’s worth paying attention to – not as gossip but because it’s a snapshot of one of the biggest debates facing the left.

Today Labour announced university tuition fees will be abolished as early as this autumn if it wins the election.

However, there was reportedly a clash over whether this should be the manifesto’s lead education policy: while John McDonnell, the shadow chancellor, prioritised free tuition, Angela Rayner, the shadow education secretary, is said to have championed more spending on early years education.

There are good reasons for abolishing tuition fees – or at least drastically reducing and capping them. As recent years have seen the cost of a degree soar, twentysomethings are now saddled with upwards of £50,000 worth of debt, only made worse by the fact that they’re often graduating to low-paid or insecure jobs and a housing market that shuts them out of finding their own home.

It isn’t hard to see why there’s a fear that high fees put off poorer pupils. A free higher education system also holds strong symbolic value: university isn’t a commodity, a calculation only the elite can afford, but a collective enterprise that benefits society.

But the evidence suggests free tuition isn’t the progressive measure it appears. Look at what it means for young people from low-income families: while there are signs that the number of state school pupils going on to higher education has dropped since fees rose to £9,000, the universities admissions service, Ucas, states that in 2015 application rates for 18-year-olds living in disadvantaged areas in all countries of the UK actually “increased to the highest levels recorded”. Though still persistent, the university inequality gap is falling: in 2006 advantaged UK 18-year-olds were 3.7 times more likely to apply than disadvantaged 18-year-olds; by 2014, when the higher fees had come into force, that ratio had fallen to 2.4.

Scotland – where there are no tuition fees – is a warning that if fees are scrapped, wealthier students often benefit, especially if it’s not backed up with means-tested grants. A cross-nation study into student funding across the UK (for the Centre for Research in Education Inclusion, and the Economic and Social Research Council) in 2014 found that while free university tuition coupled with cuts in grants to lower-earning students means middle-class families in Scotland are £20m a year better off, the overall costs to poorer students have gone up by at least £32m annually.

The lead researcher, Lucy Hunter Blackburn, doesn’t hold back: “Free tuition in Scotland is the perfect middle-class, feel-good policy. It’s superficially universal, but in fact it benefits the better-off most, and is funded by pushing the poorest students further and further into debt.”

If politicians want to build an education system based on equality – and with it, prevent a family’s wealth from determining the life chances a young person has – rather than focusing on university fees, they would do better start over a decade earlier: all the way back to early years education. Inequality takes root young. Study after study shows that not only does poverty constrain the way a child’s brain develops, but by the time we hit our teens – and often even younger – average-ability middle-class kids are even pulling ahead of the poor kids at school who have higher ability.

Pushing funding towards early years education, as Rayner wanted, provides a hugely effective way to tackle this: providing anything from nutrition advice for parents to high quality preschool teaching. But under the Tories, early intervention services like youth and children’s centres have seen their funding cut by 31% between 2010 and 2016, while one in 10 maintained nurseries – relied on primarily by children from deprived backgrounds – and a third of Sure Start children’s centres closed their doors.

Add that to a climate in which squeezed wages and cuts to in-work benefits are pushing more children into hardship – by 2022, the Institute for Fiscal Studies estimates, more than 5 million children will be in poverty – and we’re bang in the middle of a crisis in which disadvantaged kids are going into primary school with their already unequal life chances shrinking even further.

Paediatricians warned this month just how grim this is getting: children with stunted growth because parents can’t afford healthy food; respiratory illnesses caused by cold, damp housing; mental health from financial stress. About 400,000 children in this country currently don’t have a bed of their own to sleep in. Without even a good night’s sleep, a healthy breakfast, or a quiet room to do homework in, there’s a growing class of children who – purely based on the wealth of their family – are blocked out of flourishing in school.

Yes, a progressive education system would allow young people to get the benefits of university without vast debt – and protect measures like maintenance grants or the disabled student allowance, to help marginalised students afford a degree. Yet without putting significantly more funding towards quality early education, millions of disadvantaged children – unable to get the grades – will never even have the chance to apply.

A bid to push £10bn a year towards children’s centres and nurseries won’t make big headlines, or inspire marching in the street. But it would be an education policy that genuinely makes strides to equality, one that could actually address inequality at its root. As Britain enters an era of large-scale child poverty, this is a conversation we need to be having. Labour pushing its education funding to early years could have been the most radical move of all.