The liberal commentariat emitted a collective whoop when David Cameron finished his speech to the Conservative party conference last week. "David Cameron is the new leader of the British left," gushed Dan Hodges in the Telegraph. Meanwhile the Financial Times, caught up in the exhilaration of the moment, made the fantastical claim that "it takes an Old Etonian to extol equality of opportunity as confidently as David Cameron did".

Those quotes are fairly typical. They were made by centrist commentators and thus demonstrate where the centre-ground in politics currently sits – and how David Cameron has managed to straddle it.

Coming in for particular praise were the sections of Cameron's speech which dealt with 'equality'. In this respect his rhetoric struck all the right notes. Thus he said: "Opportunity doesn't mean much to a British Muslim if he walks down the street and is abused for his faith.

"Opportunity doesn't mean much to a gay person rejected from a job because of the person they love. It doesn't mean much to a disabled person prevented from doing what they're good at because of who they are. I'm a dad of two daughters – opportunity won't mean anything to them if they grow up in a country where they get paid less because of their gender rather than how good they are at their work."

That a Conservative prime minister can deploy anecdotes like this at Tory conference without being driven from the stage by a chorus of boos is testament to how far the Tories have come under David Cameron. In the 10 years since he became party leader, Cameron has gone some distance to ridding the Conservatives of the 'nasty party' tag of left-wing caricature (ironically it is Home Secretary Theresa May, who originally coined the term, who now carries that flame).

Yet for all the hand-wringing by Cameron about the persistence of racism, sexism and homophobia in Britain – none of which ought to be sniffed at – missing from his passionate plea was any acknowledgement of the inequalities perpetuated by class. The prime minister riffed on equality, but it was of a very parochial and conservative kind.

Britain remains a society riddled by class inequality. According to a 2013 study by the London School of Economics, a disproportionately large number of places at Oxford are taken up by people with Norman Conquest surnames, such as Baskerville, Darcy, Mandeville and Montgomery. Meanwhile just one in 10 of those who attend either Oxford or Cambridge had free school meals as children, compared with a fifth in Britain as a whole. And only around a fifth of the poorest youngsters go to university at all, compared to 57% of the richest.

During his speech, David Cameron gave some striking examples of the shameful inequalities that persist in Britain. But where was the talented child with no place to do his homework because his parents cannot afford to heat their home? Where was the young girl educated at a state school but usurped by the child of a private school system in the university admissions process? Where, in other words, was the class politics?

The Conservative Party shies away from talking about class for obvious reasons. But class analysis, at one time the raison d'etre of the socialist movement, has steadily been losing ground on the left too. Hence the widespread acceptance that Cameron's speech was in some sense 'progressive'. Whereas at one time the struggle for a more classless society was paramount on the left, liberation today is largely confined to the (legitimate) grievances of women, gays and ethnic minorities.

An ungenerous interpretation of left-wing history would say that an ungrateful British working class, seduced by Thatcherism and New Labour, failed to live up to left-wing expectations, so the left moved on like a bored lover to those deemed more deserving of its support. To paraphrase the much-paraphrased Bertolt Brecht, British workers have shown little interest in overthrowing capitalism and thus were dissolved in favour of a new proletariat.

Yet in reality the left's adoption of identity politics was an understandable response to the injustices of the 20th century. Sexism, racism and homophobia pervaded the socialist movement, with women in particular solemnly informed by male comrades that their demands for equality could wait until capitalism had been overthrown. For the ageing male dinosaurs who dominated the movement, it never was an 'appropriate time'. "After the revolution things will be put right," women were routinely told. And like the apocalyptic forecasts of Armageddon made by crackpot Christian sects, the revolution never did quite materialise.

The left, then, badly required an injection of much-maligned identity politics. However, as the African-American academic Adolph L Reed has written, it is important not to exaggerate the partial nature of the victories that come with greater inclusiveness. The equality preached by a Tory prime minister is now indistinguishable from that advocated by the liberal left; and if you can't find anybody who is against an idea there is a good chance that idea doesn't go far enough.

So David Cameron's equality ought not to be the version advocated by the left. His is an equality in which, among other things, corporations project a multicultural image while raking in profits on the back of exploited labour. Under Cameron's celebrated liberal orthodoxy, embraced by swathes of centre-left and centre-right opinion, bosses will still be paid 130 times more than the average worker; the difference today is that the boss will occasionally have a darker skin or be missing the Y chromosome. There is no denying that this is progress on what came before. But it is an equality with which free market capitalism, and all its obscene discrepancies of wealth and opportunity, can co-exist quite comfortably.