When I turned up at university in the mid-1970s, not a word was said about the pit boots, bunnet and whippet.

Perhaps everyone at Edinburgh was too polite to mention that something might be amiss (that was a new word). Or perhaps I was too busy being upwardly mobile to notice.

As a matter of grisly fact, the things I actually wore would have made the worst bunnet look like a touch of style. The part about politeness was true, however. There were few enough working-class heads bobbing in a sea of the differently vowelled, but the oddity was never acknowledged. That would have been rude.

It was accepted, in any case, that if you had got there fair and square (more or less) you had earned your spot on the mobility escalator. That always went upwards, for some reason. It had no other point.

To this day, no politician boasts of his undying commitment to social mobility while admitting that it could, or rather should, mean losses as well as gains. In a society that allowed for truly free movement between the classes, there would be descents as well as ascents, status lost and – for whatever it's worth – status gained. This detail is always overlooked.

A case in point would be the speech given in Glasgow on Monday by Alan Milburn, the former Labour minister who chairs the Coalition's Social Mobility and Child Poverty Commission. As we reported, he produced grim if familiar numbers to illustrate the extent of Scotland's social divide. Seventeen per cent of our children live in poverty, a number that's expected to rise to 23%. Only one in 40 pupils from the most deprived households achieves three As at Higher. Only 27% of students from working-class backgrounds are in higher education. Meanwhile, with just 5% of the young in private schools, one-third of Scotland's most senior civil servants were educated privately. Milburn argued that if we want social progress, this state of affairs has to change.

Who opposes progress? Put it this way: how enthusiastic would those mandarins and their families be about spending fortunes on private schooling only to see 95% of the best jobs in public life go to the state educated? Might not their stake in the status quo be one reason why, as Milburn observed, "The link between demography and destiny has remained stubbornly unbroken"?

Such is the dark little secret behind all the passionate talk of social mobility. It might just be why in 2011 only a pitiful 2.7% of St Andrews University students were from Scotland's most deprived communities. It might be why, according to a now notorious statistic, the young of Surrey claim almost as many Oxbridge places as their counterparts in Wales and north-eastern England combined.

When you have a guaranteed spot on the escalator, talk of social progress becomes little more than lip service. Self-interest, never confessed, is a more powerful commandment than anyone's burning desire to end an injustice. If it suits, you can take cover, meanwhile, behind the idea of attainment.

The public schools bag more places at the best universities because their pupils attain the best exam results. What could be fairer? The reality has nothing to do with fairness. It rests on a self-perpetuating system in which money buys education. Education at a "good university" in turn guarantees status and the jobs that provide the cash to buy education for the next generation. So the majority are excluded, decade after decade.

Everyone knows this is how Britain works. The parents who "go private" do not do so to provide education for its own sake. Those who covet an Oxbridge PPE don't do so in the serious expectation that anything beyond the rudimentary can be learned about politics, philosophy and economics in three years. The effect is to make a nonsense of talk about social mobility.

Improved education for all was supposed to be the antidote, once upon a time. Those who meant to extend democracy through schools and universities did not count on the tenacity of vested interests, but the idea was right. For a while, it almost worked. You could go from one of those "bog-standard comprehensives" to a university like Edinburgh and be part of a growing minority. Now the doors are closing again.

Some universities have tried hard to "widen access". Glasgow is one; my own another. The Scottish Funding Council is doing its bit, with a £40 million scheme that will see 19 universities provide 727 places for people from tough backgrounds. Fingers crossed, but it counts as no more than a start. According to the Centre for Educational Sociology at Edinburgh, in any case, nothing else seems to be working.

The working class are no more numerous in higher education thanks to the abolition of the graduate endowment fee than they were in the mid-1990s. The proportions of students from all backgrounds are, in fact, pretty much as they were. Equally, the deceitful claim that the fees demanded in England would fund wider access stands exposed. There, too, things are "static", save for evidence that – lo and behold – the private schools have a tighter grip on Oxbridge than before.

Let's pretend, then, that there are a few people on the escalator who care about Mr Milburn's social progress. David Raffe, professor of sociology of education at Edinburgh, has said: "There is little evidence of change in social class inequalities either in access to higher education as a whole or in entry to higher-status universities, including in Scotland where fees for full-time students have been abolished." So what's to be done?

You cannot, in seriousness, enforce equality of access through quotas or the like if educational standards are to be maintained. It would be counter-productive, to say the least. That means the quality of applicants from poorer backgrounds must be improved. That means their schools must be improved. But if those schools are in difficulties that is because, nine times out of 10, they have to cope with the innumerable problems caused by those same "poorer backgrounds". If those with spots on the escalator won't give up their privileges, education alone can never be the instrument of change. It might be worth wondering why a tiny minority can exert such a hold over a system that is supposed, in theory, to serve all, but that's another argument. What we know is that it is no accident.

Scotland has a different kind of problem. After all, the abolition of the graduate endowment was supposed to change everything. The research from Edinburgh, surveying the years between 1996 and 2010, says it has done no such thing. Not even the economic crash can be blamed for causing working-class youngsters to decide higher education isn't worth the bother.

Is every playing field level? Not for as long as the cash ethos of the public schools survives. No government will attempt the abolition of those places, more's the pity. That being so, a Scottish administration has no choice but to attempt to balance the scales. Educational provision has to be improved to counter the effects of economic inequality.

That's a huge task, obviously enough. But if nothing is done, social mobility will continue to be a myth.