Every prime minister chooses when and where they want to expend political capital on effecting change. Unlike a departmental minister, they have some freedom about taking a risk with their own popularity and the fortunes of their party.

Andy Burnham has made his choice: social care. Driven in part by his personal experiences of his father suffering from dementia, and in part by an ambition to make the system work that dates back to his time as health secretary in the Brown administration, he has pointed the way.

By way of a preparatory mea culpa, he has conceded that the politicians of his generation are guilty of a “major dereliction of duty” in this field – and that their collective performance is “shameful”. So he wants to do something, at last, to make amends.

But… what? There is no doubting Mr Burnham’s sincerity, nor his analysis of the mess things are in. It is a dispiriting vista. Since the Royal Commission on Care was set up in 1999, there have been 22 such inquiries and sets of recommendations. All have had limited impact, at best. When they’ve finally made it into government policy or an election manifesto, they have invariably been weakened or abandoned in the face of the backlash – against a “death tax” and “dementia tax” – and accusations of “broken promises”.

There is universal agreement on the scale of the problem, and a total lack of consensus on its solution, even when the task of policy-making has been outsourced to a battalion of respected non-partisan experts over the last three decades. The “Big Conversation” that Mr Burnham now wants does have the quality of a man studiously reinventing the wheel.

Mr Burnham plainly thinks it will be different this time – or at least could be. Instead of “point-scoring”, he seeks agreement, a common theme of his, and he has invited the main opposition party leaders to engage in talks. In fact, Conservative and Liberal Democrat shadow ministers have been quietly discussing matters since Sir Keir Starmer began his own attempt at reform two years ago, so some modest progress may already have been made.

The prime minister is right to try and quicken the pace, and to bring forward the work of the Casey Commission that Sir Keir set up, so that a framework for reform can be achieved much more quickly. The Greens, Reform UK and the unions would also like to be consulted; and if the idea of a National Care Service is ever to work, then there must be some role for the private sector.

It is difficult to be optimistic about the prospects for much agreement among such a wide variety of vested interests, some heavily invested in the failure of the initiative. Too many red lines are already being laid down. Kemi Badenoch has already ruled out any tax rises and demanded reform be “fair to those who have made provisions and saved up over the course of their lives”.

Nigel Farage has dismissed the whole project as a “uniparty” plot, while the TUC favours levies on wealth and assets. Even Mr Burnham is bringing certain assumptions to the table, favouring the public and charity sector over private companies, and insisting on an NHS-style principle of “free at the point of access”.

Scotland, meanwhile, has developed its own model of reform, separating publicly funded nursing care from the cost of accommodation, under a means-tested system.

It’s probably unwise to enter into such talks assuming failure, but it may well be their fate – Ms Badenoch has said as much – which will still leave Mr Burnham, possibly with Sir Ed Davey for company, looking for some answers.

In the short term, and if Mr Burnham actually wants to get anything done, he could do worse than return to the eminently pragmatic answers put forward by the Dilnot Commission in 2011, suitably updated. Obviously, the British population is older and sicker than it was then, but the burden-sharing approach still has much to commend it.

The idea was to cap any costs borne by an individual over their lifetime for care to around £35,000 (equivalent to £55,000 today). In addition, the state would be liable for “extreme” costs over that cap. So savings and the family home won’t ever be completely wiped out, exactly the outcome that should appeal to Ms Badenoch and her supporters.

For people in residential care, about £10,000 a year would be charged (up to the cap); for poorer people in care, such fees would be waived under a means-test. The cost to the taxpayer would be in the low-single billions, rather than the £18bn that a copy of the NHS would cost. Taxes might thus have to rise only modestly, if at all.

Indeed, if the cross-party consensus does fail, Mr Burnham should just get on with the job of fixing the system, challenging the other parties to present a better, fairer plan. He can rightly point to their unwillingness to work together – a neat little trap Ms Badenoch and Mr Farage have already tripped into. They will be seen to have no better answers of their own.

Yet, at the moment, Mr Burnham seems a little timid, pleading in his last television interview that it takes time, he can’t put a date on it and would prefer lots of collaboration. That is not how great leaders fix problems, change the political weather and win elections.

We’ve been dithering and delaying too long, and now, with a landslide Commons majority, is the time for action. In footballing terms, Mr Burnham has the ball at his feet. He can do this.