I've always had a soft spot for Sir John Major.

His prime ministerial activities were the first I followed with any degree of interest, while I appreciated the fact he wasn't your typical Conservative. State-educated, from an unconventional (rather than working-class) background and without a university degree, he was a poster-boy for post-war British social mobility.

He's also rationed his pronouncements since swapping Downing Street for the Oval 16 years ago, which means his interventions pack added punch. Comments to grassroots Tory activists recently were a case in point. That the "upper echelons" of public life were dominated by men and women either privately educated or drawn from the "affluent middle class" was something Sir John said he found "truly shocking".

This, of course, is demonstrably true: since the 1980s social mobility in the UK has worsened rather than improved. Sir John, however, undermined himself by blaming the previous Labour government, not to mention skirting over his membership of the administration (Mrs Thatcher's) that did the most damage. That social mobility flat-lined under Mr Major's own premiership wasn't acknowledged.

For the past week the former PM's speech has been the gift that keeps on giving: Michael Gove, who made similar remarks a few years ago, said Sir John was spot on; David Cameron blamed lack of "aspiration" among state school pupils (a comment which was either naïve or insulting), while William Hague said social mobility had declined along with the quality of comprehensive education, something he claimed Mr Gove was attempting to reverse via "free schools".

Worsening social mobility is one of those phenomena everyone agrees is a Bad Thing, yet the unanimity evaporates when it comes to actually doing anything about it.

The most sensible response to Sir John's comments came from Alan Milburn, the former Labour Cabinet minister and the Coalition's social mobility adviser.

"The shocking lack of social mobility is entrenched in British society," he said. "There is a glass ceiling in British society - and more and more people are hitting it."

Depressingly, the mooted Milburn manifesto, which includes compelling schools and universities to ensure their doors are open to a wider pool of talent and ending unpaid professional internships, stands little chance of being implemented. In a political culture obsessed with winning over the all-important Middle England (and its Scottish equivalent), such initiatives will most likely be filed under "too risky".

Strikingly, Sir John's remarks also attracted relatively little response north of the border. Even Yes Scotland, which rarely wastes an opportunity to point out the UK is the fourth most unequal country in the Western world, was silent.

Indeed, there's a tendency in Scotland to view a lack of social mobility as an English problem. The constant focus on Oxbridge admissions (50% from private schools) perhaps encourages this, though the record of St Andrews (40%) and Edinburgh (30%) is little better. Even my alma mater, Aberdeen, draws 20% of its intake from the independent sector, in a part of the UK where less than 5% of pupils are privately educated (marginally lower than the UK's 7%); in Edinburgh that figure gallops up to one in four. Curiously, this is little mentioned, particularly by an educational establishment that prefers unhelpful mythology ("Scottish education is the best in the world") to uncomfortable truths.

And while it is true Scottish politics is considerably more meritocratic than in the UK as a whole (although even Holyrood has a sprinkling of Old Etonians), the same could not be said of its professions - particularly law - which often resemble what Ian Bell called "reserved occupations" for the products of public schools.

As a recent Stirling University survey demonstrated, inequality in Scotland is only lower than in the rest of the UK because of particularly high levels of inequality in London. Over the past decade Scotland's top 1% of earners - around 25,000 people - have come to earn around 20 times as much as those on the lowest incomes, a trend seemingly uninterrupted by the 2007/08 financial crisis.

Such gaps between rich and poor, privately and publicly educated are, in truth, largely resistant to constitutional change. As the Stirling study shows, more than 14 years of devolution has done little to increase social mobility or decrease inequality in Scotland, so it's tempting to conclude independence would not succeed where devolution has failed.

As Gerry Hassan and James Mitchell conclude in their engaging book After Independence: "Beyond the Yes/No certainties of each camp, the most important divide in Scottish public life is actually between those with power, privilege and status, and those groups and individuals without these characteristics."

The two crucial components of any genuinely meaningful reform, political will and creative policy making, are sadly lacking in modern Scotland. While the Scottish Government's Curriculum for Excellence should be applauded for acknowledging something had to change in the state sector, Education Secretary Michael Russell never even mentions Scotland's 53 private schools, yet they are the elephant in the room, and a large one at that.

Few, of course, are suggesting abolition; an extreme policy response even Clement Attlee's post-war government rejected. While private schools demonstrably reduce social mobility, they are also self-evidently centres of educational excellence. But mitigation is possible, as the politely coercive actions of the Office of the Scottish Charity Regulator (OSCR) demonstrate; over the past few years it has been nudging Scotland's independent schools into doing more to justify their charitable status which, after all, saves them rather a lot of money.

But it still isn't enough. The OSCR could insist a steadily higher proportion of pupils in private schools should be funded and drawn from lower socio-economic groups, while the assisted places scheme - abolished by an untypically dogmatic New Labour government in 1997 - could be reintroduced as another piece of affirmative action. More independent schools could share expertise and facilities with comprehensives, as proposed by Labour peer Lord Adonis.

Action should not stop at school level, for access to the best universities remains another glass ceiling in terms of social mobility (even with free tuition). Why not reserve a certain number of places for the brightest pupils from each state school at Russell Group institutions? After all, such places already exist - to all intents and purposes - for those from the independent sector.

"Our education system should help children out of the circumstances in which they were born," Sir John said, "not lock them into the circumstances in which they were born." A noble - indeed necessary - aim in Scotland as much as the rest of the UK, and one, as Sir John noted, that isn't going to happen "magically". It needs bold steps rather than rhetorical half measures, and lots of them.