News that Oxford and Cambridge cut the number of UK undergraduates over the past 10 years – when many other universities increased theirs – and recruited more international students instead highlights the exceptionalism of England’s two oldest but oddest universities.

They are the commanding heights of our higher education system, and close to the top of global university league tables. Their dreaming spires and sheltered quadrangles and courts are part of England’s timeless heritage. Both their success and antiquity make it difficult to ask the question whether they are still fit for purpose in the 21st century.

Most of their students still come from socially privileged groups, with a grossly disproportionate number from independent schools, despite the best efforts of grassroots access and outreach staff. Many Oxbridge students aspire to join the nation’s elites, whether public service or hyper capitalist. They help perpetuate England’s “chumocracy”, whose deficiencies have been on such cruel display in the Brexit negotiations compared with rival European elites trained, for example, in France’s grandes écoles.

The reduction in the number of UK undergraduates has made it even more of a challenge for both universities to recruit students from more disadvantaged – or, maybe, just more normal – social groups. Widening participation without increasing participation is difficult, because more-deprived applicants then appear to be displacing those who are more privileged (with their vociferous and influential parents).

Independent schools watch like hawks for any sign of “social engineering” – in other words, levelling the admissions playing field. Getting their pupils into Oxbridge, after all, is their USP. They, and the more successful state schools, are adept at grinding out the top grades, which Oxford and Cambridge, in thrall to an “excellence” that gives little weight to the gross inequalities of school education (not to mention the English class system), have come to expect.

Yet at the same time, they are without question world-class research universities. The roll call of past Oxbridge scientists and scholars exudes eminence. These two universities continue to play a leading role in research in most, although not all, disciplines, not only in the UK but worldwide.

Oxbridge sometimes resembles a pantomime horse – undergraduate finishing schools fighting with world-class research in the same sack. Efforts to adjust the balance between the two, to the advantage of the latter, have been too tentative. New graduate colleges have failed to challenge the dominance of the original undergraduate colleges. The recruitment of more postgraduates, often international students, is a step in the right direction. But, in the absence of tougher policies on fair access, this rationing of undergraduate places makes entry even more competitive.

Then there are governance and constitutional weaknesses. Put bluntly, colleges, especially at Oxford, have too much power and are difficult to reform. Some have barely escaped the shadows of Barchester Towers – if the current wrangle between Christ Church and its dean is anything to go by.

Resources remain unevenly distributed. Super-rich colleges enjoying medieval wealth coexist with underfunded university departments. College fellows with parson-like freeholds (let’s forget the wine cellars for the moment) rub shoulders with researchers on precarious contracts. Efforts to redistribute wealth have achieved little real change.

The simple truth is that Oxford and Cambridge are too good and too important to be allowed to remain unreformed. They need to do more – much more – on fair access if they are to reflect, even approximately, modern Britain. Quotas for students from less-privileged backgrounds are essential, even if they squeeze out some of the gilded youth. They also need to use their resources more rationally by taxing excessive college wealth if they are to continue to compete with the likes of Harvard – and, in the future, the Far East.