It is still just about possible, with a sustained deployment of the imagination, to conceive of a set of circumstances in which Theresa May might still be able to regard her Brexit policy as a success. If, for instance, she agrees in Brussels on Thursday on the conditional short extension offered by Donald Tusk on Wednesday ; if she gets her third “meaningful vote” motion past the Speaker next week; if her MPs take Tusk’s offer seriously; if she proves better at persuading enough of them to back her deal than in the past, then – just possibly – it may all seem to have been worthwhile. Or at least it may do so for about 48 hours, until the leadership challenge begins and the policy conflicts of the next – and longer – phase of Brexit negotiations with the EU start to split her party yet again.

That imagined moment, though, has now become such a vanishingly small possibility that the final verdict on May’s Brexit policy must now be of its conclusive and overwhelming failure. Her Brexit policy was always to get Britain out by 29 March with a deal that postponed all the major policy implications, on the basis of Tory support, buttressed only by the Democratic Unionist party and Labour’s handful of anti-Europeans. It was always going to be a close-run thing. But it is not going to happen now.

The date has been missed because of the extension letter she sent to Brussels on Wednesday . And her diehard anti-Europeans will not be enough to get her over the line if there is a third meaningful vote next week. Indeed, May’s promise on Wednesday, in effect, to resign if Brexit is delayed beyond 30 June was almost an invitation to parliament, her party and the EU to scheme her departure by supporting a longer delay.

May’s performance at prime minister’s questions was that of a beaten leader. She was, finally, cornered. She knows now that she will lose. She has no other cards left that she is willing to play. In the past, she has often wagged her finger at MPs as she told them to get into line behind her deal. Now there is a new vehemence in her admonishments, precisely because she cannot win. “This house has indulged itself on Europe for too long,” she complained , as if she was summoning her own inner Oliver Cromwell. There are many reasons why he was a greater leader than May, but, in the end, when parliament defied Cromwell, he possessed something May doesn’t. Troops.

May’s Brexit policy – hard not soft – was always misguided. Now it is also a failure. Talleyrand’s reaction to Napoleon’s execution of one of his enemies applies here: it is worse than a crime, it is a mistake. Nearly three years ago, May had the scope and the authority to craft a Brexit strategy that would not have failed. She could have targeted a withdrawal process that would work for some remainers as well as for leavers. But she set her face against that opportunity. She elevated her chosen policy above her political common sense.

This is almost always a mistake. The relevant comparison isn’t really with Cromwell or Napoleon long ago, although they, like May, were also attempting to navigate a revolutionary movement towards greater political stability, as she is. The comparison is more recent. It is with Tony Blair, who became convinced that sticking with Washington over Iraq was the right course, and who pursued the policy so single-mindedly and to such an extent that he could not see that the policy was failing, that there were other options, and that it might destroy his reputation.

‘May’s performance at prime minister’s questions on Wednesday was that of a beaten leader. She was, finally, cornered. She knows now that she will lose.’ Photograph: Jessica Taylor/EPA

With exquisite timing, May is now about to witness something that Blair also experienced as his policy ran aground. In 2003, Blair watched a million people march past Downing Street pleading with him not to join the invasion. He concluded, catastrophically, that they were a million of the misguided. Sixteen years later, May could be about to have the selfsame experience, with Saturday’s Put it to The People march perhaps also drawing a million people back on to the streets.

There is no iron law about the political significance and impact of large street demonstrations. The 21st-century experience, however, is that British politicians who ignore very large and peaceful protests are taking a risk. Parties and unions cannot put a million people on the streets in the way they once could. Today, it’s the issue that mobilises. It would be no surprise if May, like Blair before her, dismissed this weekend’s march as of no consequence. But she would be wise to learn from his mistake.

Blair’s political error on Iraq was not to recognise his own policy failure. When the United Nations did not back military action, he should have accepted it. He should have told the public that he had been willing to try everything to win support for invasion, but that Britain could not now proceed. He would have been right and wildly popular. Instead, from that moment to this, he was pigeonholed as delusional.

May’s political mistake on Brexit is very similar. When she applied for an article 50 extension this week, she should have recognised the verdict. The old policy is dead. Even now, she could still turn to the country and remind us that she has done everything she conceivably could to get the Brexit she preferred. With regret, she could say, we must now go in a different direction, perhaps leaving the EU but preserving the benefits from the single market and customs alignment. It wouldn’t please all the marchers any more than it would persuade the European Research Group. But it would be pretty popular, all the same.

The overarching question that faces British politics is not whether May’s Brexit policy will succeed – because it has failed. Until Wednesday, the succession was a secondary issue, because the arithmetic in the House of Commons will not change. Now, though, parliament is faced by both questions. The country needs a new Brexit policy, and it may soon need a new leader to try to implement it.

• Martin Kettle is a Guardian columnist