For two long years, Theresa May always insisted that Britain will leave the European Union on 29 March. So determined has she been to deliver Brexit by this self-imposed deadline that she has even had the date written into law. This week, she made an uncharacteristic bow to political reality and accepted that it may not happen after all. This was a pivotal concession and a watershed moment in Britain’s argument about Brexit.

Mrs May’s statement on Tuesday should be read with care. It was hedged with conditions. She continues to pursue her original timetable. Her own deal – possibly slightly tweaked from the version that MPs threw out in January – would have to be defeated on 12 March for any delay to Brexit to be triggered. Parliament would also have to vote – again – against leaving the EU with no deal. If both these things happen, and both currently appear likely, the government will then put forward a motion to MPs for “a short limited extension” to the article 50 process. Mrs May made it clear that she will oppose any extension beyond the end of June 2019.

The prime minister’s willingness to allow an extension is a welcome development. It had to be forced out of her by some equally welcome tough action by a group of cabinet ministers. But the terms in which she might approach any extension of article 50 remain unrealistic. She needs to be much clearer – with herself and the country – what she would use the extension for.

Based on what she said this week, she appears to hope she can use an extension to get yet another variant of her existing deal over the line. But there is very little negotiating space for that. The EU is not willing to change the withdrawal agreement. Her hardline leavers are not willing to bend the knee and accept it. If she sticks to the strategy of making meaningless tweaks to her deal to make it acceptable to hardline leavers, we could arrive at the end of June in exactly the same mess as now. Mrs May therefore needs to recognise another truth. A delay will not mean business as usual. It will mean a rethink. This is why the agriculture minister, George Eustice, resigned over the issue on Thursday.

Such a rethink could take several different forms. She could try for a softer Brexit deal, perhaps involving customs and single market alignment along European Free Trade Association terms. She could commit to lock Britain into a customs union. She could offer an early general election in return for votes for a changed deal; or accept a second referendum in exchange for support for her deal in the way that Labour appears to be contemplating. She could embark on a deliberative democratic exercise to seek out public consensus.

Any of these, or any combination, would involve reaching out to different interests and large changes of approach. That process necessitates new thinking. Mrs May is not a natural new thinker. But there is no alternative to trying. Her existing Brexit policy has failed. It will have to be replaced by a better policy that commands wider support. This will need time. Not unlimited time. But probably more than three months, especially if the outcome is ultimately to be put to the voters in an election or another referendum. All this suggests an extension of at least six months, and maybe more. The lesson of the current impasse is precisely that we cannot again let the clock prevent Britain from sorting Brexit out.

Mrs May told MPs this week that no extension can go beyond June because that would mean the UK taking part in the European parliament elections in late May. That would be a problem if the Brexit outcome was settled. But it isn’t – that’s the whole point. Mrs May should get her priorities straight. The elections are awkwardly timed but it is silly not to take part. The priority of getting this national emergency resolved properly is worth the price of having British MEPs in place while it is being sorted.

Election campaigning may even be useful at a dark time. Britain is in the midst of a deep national argument about future relations with Europe. Why not open that argument to the people? For once, the European elections might actually be about Europe. Britain should certainly make clear that, if Brexit is agreed, our MEPs would withdraw. In the meantime, Britain loses nothing by taking part. Let’s vote.