As Naomi Wolf recently reminded us, young women consistently receive fewer first-class degrees than men at Oxford University, where I work, even when marking is blind. Meanwhile, Russell Group universities consistently admit a very low proportion of working-class and BME students. There are concerns that the kind of assertive declarations that win middle-class white men firsts also gain them preferment at Oxbridge interviews. There’s an inequality crisis in university admissions and assessment.

What’s to be done? We could spend the next decade, as we have the last, researching and debating this – or we could simply abolish university assessment and entry requirements.

That isn’t as crazy as it may sound. We now have a Labour leader committed to making education freely available for all and to realising aspiration collectively, rather than through competition.

The concept of the comprehensive university was seriously debated in the 1960s. I came across it 20 years ago at the University of Sussex. Many of the faculty had joined Sussex in its early days, deliberately eschewing Oxbridge to create a new type of university – one where students of all backgrounds and ages would debate and create a new society. Many students deliberately chose Sussex over Oxbridge because its innovative interdisciplinary curriculum, lack of hierarchy and emphasis on debate rather than ceremonial dinners appealed to them.

When these professors retired, they allowed us graduate students to plunder their book collections. I amassed a collection of pamphlets and books that set out how universities might apply the comprehensive ideal then taking shape in secondary schools. This wasn’t ill-founded idealism – these writers had realised that comprehensive schools were getting great results from children who a few years earlier would have been sent to secondary modern schools that couldn’t offer any education beyond the age of 15.

A comprehensive education system means not treating two universities – or 20, if we use the Russell Group – as the best

The early promise held: by the late 70s, for the first time the proportion of working-class children getting A-levels and into university was increasing. That was because of the expansion of comprehensive schools and new polytechnics and universities such as Sussex.

If we’re serious about wanting a better-educated, better-trained workforce, let’s not look to selective education for the solution. And if we want to raise opportunities for women, for black and minority ethnic people, and for the disadvantaged, then we could do worse than think about abolishing selection criteria. Why shouldn’t everyone attend the university of their choice?

Critics will say that this would end up with Oxbridge being massively oversubscribed, while post-1992 institutions would close. But even in today’s extremely stratified higher education “market”, Oxbridge is not top of the popularity stakes. In my subject, history, Bristol, Warwick and UCL consistently receive more applications per place than Oxford or Cambridge. Does this suggest low aspiration on the part of today’s young people? Given the excellent education these other institutions offer, I don’t think so. Oxbridge applicants are highly self-selecting – the small-town atmosphere, traditions and the range of subjects offered don’t appeal to everyone.

A comprehensive education system means not treating two universities – or 20, if we use the Russell Group – as “the best”. We’d have to acknowledge that there are many pedagogical innovations outside Oxbridge, often in the former polytechnics. These included courses with self-assessment and no assessment at all, as well as those emphasising collaborative learning – a key skill for today’s workforce, including academics, but one not much encouraged by our universities’ focus on competition and constant grading.

A comprehensive university system would mean redistributing funds to ensure equality. There are a bewildering array of measurements of university “success” out there: international league tables, the UK’s Research Excellence Framework, the national student satisfaction survey. Yet whichever way the cake is cut, the Russell Group always ends up with the largest slice.

Let’s stop doing that. Let’s instead acknowledge that some universities have large reserves of private wealth – Oxford is one – and insist they receive less government funding. Those institutions hardest hit by public spending cuts over the past few years will need a larger share of the money for the first few years of this new system.

And, to ensure the system is properly funded, and to encourage mature students and those from working-class backgrounds, we’ll need more cash from taxpayers – a return to paying student grants and fees from the public purse.

Jeremy Corbyn’s huge mandate suggests there is a lot of support for this. Taxpayers want to fund the NHS because we are proud of universal healthcare – not because we want to fund a handful of “elite” hospitals that tend to a tiny minority of those in need. Why shouldn’t we take that as a model for our universities?

Selina Todd is professor of modern history at Oxford University and a fellow of St Hilda’s College