A symbolic casualty of Britain’s rolling, roiling Brexit debacle is the “commemorative” 50p coin bearing the date 29 March 2019, which the government had planned to release. It was one of their more idiotic ideas to put such an item in the nation’s pockets and purses when the country is so divided and its destiny is swirled with such a dense fog of uncertainty. The coin is now as redundant as the prime minister who signed off on it. Britain won’t be leaving the EU to the deadline that Theresa May has held so sacred that she swore she would not be deflected from it on more than 100 occasions.

Her inability to meet her own date is one illustration of the failure of her broken Brexit strategy. Another is the heel-stamping manner in which she responded to this latest episode in her thick volume of humiliations. Shortly before she went to Brussels to seek a postponement to Britain’s withdrawal from the EU, the prime minister had a temper tantrum that was no prettier for being dressed up as a televised address to the nation. The inflammatory speech from Number 10, in which she sought to rouse voters against parliament by displacing the blame on to MPs, was thought “appalling” by her own chief whip. Other cabinet colleagues condemned it in much riper language.

The first error was tactical. By seeking to make villains of MPs, she was attacking the very people whose assistance she needs if she is to have any hope of getting her deal over the line in the event that she dares to make a third attempt. Whatever she does with her post-prime ministerial life, I do not recommend a career in anything that requires charm, finesse or a mastery of the arts of persuasion. The speech went down badly with both the Tory ultras whom she needed to soften and with those Labour MPs whose votes she had to attract to keep her deal alive. In fact, it went down terribly with virtually all MPs of every flavour. Leadership by ultimatum is unattractive, but can be effective when a prime minister is strong. Leadership by ultimatum is both ugly and hopeless when the prime minister is as enfeebled as Mrs May. That speech crystallised the despair with her combination of reckless obduracy and desperate devices. The consensus in her party is that she has entered the twilight zone of her premiership. One senior Tory figure, who is an astute reader of his party’s moods and has been loyally supportive of the prime minister to date, reports that it is now the “pretty universal” view of Conservative MPs that we are in the closing chapter of her premiership. The only outstanding question is whether Mrs May comprehends the severity of her plight. His concern is that the prime minister is in denial and “may not understand the realities” of her position.

The second error in that speech was simply factual.

When she invoked the “will of the people” against parliament, she spoke as if wilfully obstructive MPs were blocking a deal that is wildly popular with the public. “I am on your side,” she claimed, casting herself as the tribune of the citizenry against these dreadful parliamentarians. But the people are not on her side. The reverse is the case. Her deal is not popular. We have been sucked into this swamp of stalemate because there is no clear public mandate for any Brexit plan, but Mrs May’s deal is particularly unloved. This weekend, more than a million folk have massed in London to demand that the question be put back to the people in a fresh referendum. There are more than 4 million names on the petition seeking the revocation of Brexit. Meanwhile, Nigel Farage’s tribe, absent its leader for most of the journey, is trooping towards the capital with banners decrying Mrs May as a betrayer of Brexit. I am not aware that anyone is marching or organising a petition in favour of the prime minister’s withdrawal agreement. No one is heading for Parliament Square chanting: “What do we want? Theresa’s terrific deal!”

An analysis by the pollsters YouGov suggests that there are only two constituencies in the entire United Kingdom where there is majority support for Mrs May’s version of Brexit. That is one of the reasons why she has already gone down to two landslide defeats in the Commons. She might have cajoled more support from MPs if the public was behind her, but it is not. Rather, MPs who support her deal are in many cases doing so at the high risk of angering their party members and without the support of the majority of their voters. Those MPs voting against her deal, for their very different and often completely contradictory reasons, are closer to the mood of the public.

Mrs May did say one true thing about parliament in her Downing Street speech, which is that the Commons has been a lot clearer about what it will not have than it has been about what it will approve. MPs have twice rejected her deal and they have twice voted against a no deal. The Commons has not yet demonstrated majority support for anything else. This grim state of affairs is largely of her own engineering. She complains about a situation she has done more than anyone else to create. She has thought to bludgeon her way to a majority by reducing the question to a brutally binary choice between her deal and crashing out and fiercely resisted giving MPs a genuine opportunity to explore alternatives.

That is what the Commons now has to do and with urgency and purpose. By agreeing to an extension, the EU has displayed considerable forbearance and a commendable willingness to try to help save Britain from itself and a calamity Brexit that would hurt us and them. The EU has given Britain a breathing space. Contingent on what happens next, this could turn out to be a lengthy period for rethinking or it could be a perilously short reprieve from disaster. Britain could be back on the brink of the abyss by mid-April. There may be as little as three weeks to locate a solution to this national emergency. That makes it imperative for MPs to get on with exploring means of escape from the Brexit crisis. The government says it will this week facilitate a series of “indicative votes” to test preferences among MPs and try to discover whether there is a resolution capable of attracting majority support. If the government reneges on this promise, or comes up with a process that won’t do the job, then the Commons must insist by wresting control of parliament’s agenda from Mrs May’s desiccated grasp.

For this exercise to be worthwhile, it will have to be conducted on an unwhipped “free vote” basis. Discipline in the Tory party has so imploded and Mrs May’s authority is so shattered that attempts to whip are nigh futile anyway. Governments are traditionally allergic to free votes because they publicly expose and magnify divisions between ministers. That sort of thinking has been rendered irrelevant. The world already knows that this cabinet, which has been warring with itself for months, is utterly split. It won’t be surprising news if the chancellor votes for a different Brexit outcome to the trade secretary. The opposition parties, Labour included, should also unshackle their MPs and allow them to freely follow their own judgment without fear of intimidation.

We should not be naive about free votes. They are not a miracle cure for the Brexit palsy that some suppose. It is possible that unicorn-chasing MPs will concoct fantasy wishlists by voting for versions of Brexit that the EU has already said that it will never agree to. It is also possible that MPs will discover that there isn’t a majority to be found among them for anything if there isn’t a willingness to set aside first choices and embrace compromise. Even if this week’s voting succeeds in indicating where a majority might be found for a viable deal, that will only be a first step towards rescuing Britain. Any solution will then have to be turned into legislation. That takes time and time is short. It will be essential that MPs make a grown-up effort to try to illuminate a path out of this nightmare.

If Mrs May doesn’t like the idea of surrendering power to MPs, or if they can’t make a success of it, there is an alternative. Hand Brexit back to the people for whom both prime minister and parliament claim to speak: let the voters deliver a fresh verdict.

Both these routes to resolution require Mrs May to relinquish control of Brexit, but that is as good as gone anyway. You can’t lose what you’ve already lost.

• Andrew Rawnsley is Chief Political Commentator of the Observer