The Labour party, it turns out, still has the ability to surprise. Since that glum morning of 8 May, when the 63% who had not voted Conservative woke up to find that David Cameron had achieved an unexpected majority, the opposition away from Westminster has been re-energised. Jeremy Corbyn has shot from backbench obscurity to the favourite to become Labour leader, and there is unusual interest in the party’s direction. Many young people, long alienated from politics, have become passionate and engaged. There’s been no talk, as last time, of there being little to choose between the candidates. A yearning for plain speaking over soundbites is evident, as is a hunger for a full-throated alternative to the reinvigorated Tory government over austerity, social security and the wealth gap. There is overdue debate about the links between big business and politics, as well as about inequalities of class, race and gender. Britain’s ignored but widening generation gap has finally moved to the fore.

It has all been chaotic and confusing – but also important, because two existential questions are at stake: what is Labour for? And what does it want? And perhaps, underpinning it all, a third: does it matter if Labour disappears? Those who espouse an anybody-but-Corbyn line fall prey to the very politics that have alienated so many. They define themselves wholly negatively, by what they are against. Disparaging the insurgent’s supporters as immature, they embody the world-weary cynicism that discourages faith not only in Labour but in democracy itself.

This is no local problem. Throughout Europe, social democrats are struggling to formulate a response to austerity that can at once protect the most vulnerable and attract the broader support needed to win. The twist in Britain – outside Scotland – is that the left has not bolted. The Corbyn agenda may echo that of Syriza, Podemos or the Parti de Gauche, but it seeks its home within an established vehicle. The search for answers to urgent questions is hardly one that Labour should wish were happening elsewhere.

Under assault are two facets of the recent past. The unspun Mr Corbyn repudiates the polished “lines to take” style of politics of the Blair/Mandelson years. But he rejects New Labour on substantive grounds too – contesting the decades spent, as he sees it, triangulating and compromising with the powerful in general and the rightwing press in particular.

Mr Corbyn’s opponents counter by pointing out his own flaws. Having cheerfully defied the whip 500 times, he would lack the authority – and perhaps the will – to hold Labour’s patchwork tribe together. Many see his whole economic programme as wrong-headed, and – at the least – some of his solutions sound anachronistic. He has little to say about the new technologies on which future prosperity will depend; he has called for the reopening of long-sealed coal mines. Beneath the seemingly spontaneous activism is a determined drive by the Unite juggernaut, even while organised labour remains in historic retreat. And as he justifiably rails against the rich-poor divide, he airbrushes away other distinctions – such as the tension between consumers and producers – which can matter just as much.

Corbyn critics of another sort admire his policies but warn they will never be realised because he is unelectable. They look at the soon-to-be-redrawn parliamentary map and conclude, with some justification, that a Corbyn-led party would lose badly in 2020. Electability is not a science. There are times when a clear vision matters a lot, but self-interest is reliably important too. Things are further complicated by the detailed 2015 results, which reveal a deeply fractured electorate. Positions that win votes in Scotland or London may lose them in Middle England.

But while the quest for electability is beset by uncertainties in fluid times, it must be front and centre in any major party’s mind. Labour is not a debating society; it was founded to represent the interests of working people at the pinnacle of power. This engagement in politics, this new excitement, must be channelled towards government. The brute lesson of May is that Labour cannot get there without first winning back significant numbers of Tory voters. Mr Corbyn will not do that. Those searching for an election winner must look elsewhere.

The surprise success of the Islington MP is a reprimand to his three rivals. They have failed to inspire, coming across to too many members as a triple-headed embodiment of the well-dressed, smooth-talking Westminster class. Liz Kendall has shown courage in telling Labour audiences what they don’t want to hear about the need to win Tory votes, but she lacks experience, and too often appears to be reopening the Blair playbook. Andy Burnham is personable and has passion, but has zig-zagged too often in this campaign to be seen as a leader.

Yvette Cooper is more steadfast, consistently challenging George Osborne on economic terrain. She refuses to concede the nonsense that Labour overspending caused the crash. She would disconcert a prime minister whose clumsy and occasionally patronising tone towards women has proved a vulnerability. And, of course, after a century of male Labour leaders, a female leader would be a plus in itself, all the more so when Tom Watson is a front-runner to be deputy. Ms Cooper’s down-to-earth feminism defines her politics – she knows that those hardest hit by austerity are women – so her victory would be more than symbolic.

The new leader must confront the desiccated condition of the Labour establishment: without that, the Corbyn surge would never have happened. Whoever wins must engage with the anti-austerity arguments pulsating through the party, harness young people’s passion, and take the fight to the Tories in ways that appeal to the middle ground as well as the left. The right leader is the person who can bring both Jeremy Corbyn and Liz Kendall together in one big, progressive tent, offering enough moral common ground to transcend deep disagreements on policy. It is a formidably difficult task, but there are very many in Britain who desperately need someone to pull it off. The person best placed to do that is Yvette Cooper.