It is not often a committed advocate of comprehensive education is invited to address one of the country’s leading independent schools. But after a robust exchange at a conference between myself and the head of Westminster school, Patrick Derham, I was asked to speak to his students. Derham is one of a handful of independent school heads who grasps that something needs to change, though not quite in the way I am about to suggest to his students.

My chosen title is: What’s the problem with private education? It feels like a good time to enter the lion’s den and offer a strongly contrary view to the received wisdoms of this deeply Tory age in which the power of wealth and with it, private education, is as resonant and divisive as ever.

It was not always so. In the run-up to the 1944 Education Act, political leaders of all parties seriously debated, but ultimately rejected, the enfolding of the old public schools into the new structure of free secondary education. In the progressive 60s and 70s, schools such as Eton were considered both something of a joke and emblems of an outdated oppression, as symbolised by Lindsay Anderson’s explosive allegorical 1968 film, If. In 1973, Labour’s Roy Hattersley told prep school heads of the party’s “serious intention” to reduce and eventually abolish independent schools. It seemed logical to assume that the Berlin Wall separating private and state education would soon be dismantled.

How wrong could you be? As the Oxford historian David Kynaston, one of the most acute current critics of private education, observes: “Endless reports point to the privately educated stranglehold and the sheer disparity in life chances, but I’ve yet to see an editorial in a serious broadsheet, including the Guardian, or more than the occasional speech by a politician, that squarely confronts the issue.” He adds: “It is a sad shortfall in what is supposed to be a mature democracy.”

We have recently heard the now familiar arguments on this question from the Sutton Trust, social mobility tsar Alan Milburn, academic John Goldthorpe and Ofsted chief Sir Michael Wilshaw, while the Labour MP Dan Jarvis spoke recently of how “the daughter of a cleaner in … Kingstone, Barnsley, [should] have the same life chances as the son of a barrister in Kingston upon Thames.” Rousing stuff, but how likely is this to be achieved as long as a pupil at an expensive independent school has as much spent on their education in one year as the average UK citizen earns in total and a student at a state school outside London is educated on roughly £4,000 a year – about half a term’s fees at Westminster?

While the 700 or so pupils seat themselves in the body of Westminster Abbey, no less, to hear me, I am locked in vigorous disagreement with Derham: he is determined to put me straight on the perception of private education. (Mine is well out of date, apparently.) Certainly, Westminster is far from the Hogwarts image of old. It is more ethnically diverse and is co-educational in the sixth form. For all that, the grand public schools still offer, as in my father’s day (he was a Westminster pupil long ago) a smooth route to the top with yearly fees between £24,000 to £33,000 and the highest rates of acceptance to Oxbridge of any school in the country.

A key part of the elite product on offer, even to boarders born in Beijing, is familiarisation with the architecture and atmospheres of the English establishment, including regular services in the stained-glass glow of the abbey and the chance, today, to hear school musicians play the heart-stoppingly beautiful first movement of Bach’s Concerto for Two Violins in D minor, at which point I nearly lose my resolve.

I don’t want my speech to be just one more incantation of the injustices of our country’s shameful educational apartheid. I describe the interesting near misses of history, such as that pre-1944 possibility of reform, and more significant victories such as Finland’s astonishing decision in the 1970s to abolish all private and selective schools, thus creating one of the world’s finest and fairest education systems.

We have to find a new way through this old, old problem. Up to now, there have been two main paths to potential reform: outright abolition, as perennially called for by a few brave souls – the playwright Alan Bennett, for example – or the intermittent requests placed on the independent sector by nervous governments to do more to help state schools. Some private schools sponsor academies, although these experiments can have a whiff of patronage about them and have had very patchy results. Meanwhile, a few top private schools, such as Westminster, have backed highly selective sixth-form free schools, often at great cost to the public purse.

Serious attempts to claw back anomalous tax advantages from the independent sector have usually ended in failure. In 2011, the courts in effect returned to private schools the right to decide what constitutes “public benefit” in return for charitable status. In 2014, the then shadow education secretary Tristram Hunt was roundly criticised for daring to suggest the withdrawal of business-rate relief unless independent schools entered into a more meaningful partnerships with state schools.

But there is a third way. In Kynaston’s words, this is the “harder route to reform”: abolition of the fee-paying principle and integration of private schools into a national system. It is hard to imagine, in the current climate, any major party contemplating it. For all their supposed evangelism about educational inequality, the Tories have offered no serious challenge to private education: far from it. The whip hand only ever hovers over the underfunded and ever more demoralised state sector.

Labour, too, has shown little appetite for the fight in recent decades. But Jeremy Corbyn’s idea for a National Education Service provides a sound framework for a fundamental rethink. And political moods can change quickly.

As I tell the Westminster pupils, Finland teaches us significant lessons. Reformers need to build consensus for change across the political spectrum while resisting a return to the grammar/secondary modern divide as the price of integration. The very first thing Derham says to me as we file out: “You know you’d have to bring back selection?” But for someone like Kynaston this is “a second-order issue compared to the ending of fee-paying education.”

As for the pupils, my talk is met by what feels like a slightly tense silence. Most are rushing to classes but one boy thanks me for saying “really necessary things about the privilege in this place”. Later, both the head and chaplain send me courteous thank-you notes, indicating that I have stirred up a great deal of debate, and caused some students much “wrestling with the issue”.

What have I learned from this experience? That the next critic of private education to speak in any similar place should suggest a full-on debate, bringing in pupils from nearby state schools. Let the conversation begin.