The regularly occurring furore over admissions to Oxford and Cambridge reignited recently thanks to David Lammy. The Labour MP pointed out huge gaps in admissions when it comes to race, geography, and class, and criticised the universities’ deep bias towards those students whose parents can afford an expensive private education.

Just days later the Daily Telegraph published an inaccurate report accusing a black woman of “forcing Cambridge to drop white authors”. Much like the Mail’s “our remainer universities” report, this was so exquisitely tailored towards producing a specific kind of outrage that it seems very unlikely to have been the result of mere shoddy reporting.

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These stories about Britain’s answer to America’s Ivy League colleges, are all skirmishes in the war over who owns Oxbridge – not the buildings, but the very concept itself.

Oxbridge is generally recognised as being a home of excellence in learning, but these conflicts would not occur if that were all that was at stake. The reality is these that institutions sit at the very heart of the British establishment. An Oxbridge degree is a passport to the upper echelons of British power and public life. The myth of the Oxbridge meritocracy sees no issue in theory with giving these passports out to a wider range of people, but what rankles is how Oxford and Cambridge might themselves be changed in the process.

Oxford and Cambridge sit at the very heart of the British establishment

Lammy’s focus on access answers half a question. To the extent that Oxbridge really does filter for the best of the best and provide them with world-class teaching, letting more minorities in so that they can be passported into parliament, the media and the judiciary makes sense. But Oxbridge isn’t simply a place where students go to get taught. As one academic said to me, its “function as a positional good is more important than the knowledge, habitus and skills it delivers”.

Access itself is hardly an easy problem to solve. Dr Phil Edwards, who studied at Cambridge and now teaches at Manchester Metropolitan university, remembers being struck by the “general ambient poshness” of Cambridge but did not feel excluded by it – “for those three years, we felt it was ours”. However, there are stories of students being filtered out through self-selection rather than by the university itself turning anyone down. I heard of students with northern accents being called “pit monkeys” when they attended open days, and from disabled students who had to suspend studies because the system was too inflexible to accommodate their needs. Tautologically, if you’re a natural fit you’ll fit in easily, but we must ask if “being a natural fit for the culture of Oxford and Cambridge” and “being one of the intellectual elite of the country” are actually strict synonyms.

One academic, who wished to remain anonymous, told me: “A few years ago one of our external examiners noted that our best students were producing work as good as [that of] those at their Oxford college. They were a little bit surprised. We weren’t.” Writer and academic Sunny Singh, of London Metropolitan University, said that “aside from the ability to blag with confidence,” she saw “nothing better” about an Oxbridge graduate compared to those on her own course. She was dismissive of people who are “lauded as a great intellect because [they] can recite Kipling from memory,” without the critical understanding of why you shouldn’t do it in a temple in Myanmar.

As much as Oxford and Cambridge can produce well-educated, well-rounded students, there is no denying that this output comes with a degree of pollution, like the smoke billowing out of a coal-fired power plant. What they also seem to produce is a glut of erudite wags who are charming at dinner parties; well-spoken people who pride themselves on being able to “debate” any topic regardless of expertise or even basic knowledge. As research has shown, institutions often reward overconfidence as if it was competence, what Daniel Kahneman describes as a “substitution” error. While by no means exclusive to government circles, it is surely not hard to draw to mind examples of people whose baseless overconfidence has led to them being put in charge of situations they had no idea how to actually manage. These qualities seem like negatives rather than positives for the job of governance (as distinct from getting elected), but they are nurtured and rewarded by the culture of our elite universities.

If the Bullingdon Club boys promoted above their ability are the obvious pollutants, a congealed fatberg of privilege clogging the arteries of government, the broader hidden pollution of Oxbridge’s domination of public life is the creation of an intellectual monoculture. Such institutional biases are emergent rather than explicitly imposed – a result of internal cultures that mistake the range of diversity within an organisation for the full range of possible diversity in the wider world. An attitude of “what we don’t know isn’t worth knowing” can lead to organisations careering off the unseen cliff edge of their own collective ignorance.

Oxbridge colleges may offer excellent teaching, but they do so in a shielded environment. Leafy lawns, rowing rivalries, and centuries of history are part of the appeal. As someone who used to teach in Oxbridge told me, even academics can be tripped up by the idiosyncratic systems. “It’s not that people won’t tell you if you ask them,” they said, “it’s rather that if you’re not someone brought up in the system … you identify yourself as the outsider because you constantly have to ask ‘what does this mean’ ‘how do I do that’ etc. It’s very alienating, without even being a deliberate act.”

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Adapting to this system can be impossible for some, not through lack of “natural talent” but simply because the system can’t be adjusted sufficiently for the needs of, say, single parents or poor students living at home, without losing much of what makes it special. It seems absolutely unfair to say that people in the admissions system aren’t working hard and trying their very best to reform – everything I’ve seen suggests the exact opposite. But Oxbridge is Oxbridge fundamentally because of its character and traditions, not because of its exam results, and if those traditions are the problem, then what?

In some ways thinking about Oxbridge as a coal-power plant is instructive. If it can’t be reformed, if something singular about it means it will always spew black smoke over the countryside, then the externalities need to be dealt with in a clear-eyed manner. So should Oxbridge be mothballed in the same way as we would close down any other industry at the end of its useful life?

Or is there a way we could practise the equivalent of carbon capture and storage, attempting to retain the benefits while cutting down on the toxic emissions? Prof David Andress of the University of Portsmouth, has suggested a novel approach. In his view Oxford and Cambridge should stop taking undergraduate students altogether, breaking the conveyor belt from private school to cushy establishment sinecure midway.

He suggests instead that they become “postgraduate centres of excellence,” open to UK and international graduates, “with appropriate financial support”.

Some might suggest this would simply push the problem on to other Russell Group universities, which have their own problems. But these problems seem more tractable at that level, and there are more of them, which would help dilute the issue. In addition, taking Oxbridge out of the equation would mean the other Russell Group universities could no longer hide their own issues with student recruitment behind the regular furores over the big two.

Such proposals are almost absurdly radical within our current Overton window of acceptable thought, but that is precisely the point. If marginal reforms seem unlikely to work, then we need to consider the big moves. If clinging to tradition ends up damaging our society, surely it is time to let those traditions go.

• Phil McDuff writes on economics and social policy