MPs’ reflex reaction to the new six-month Brexit delay was a very human one. They decided to take a break. They have awarded themselves an Easter chill-out, starting tomorrow, following the most punishing peacetime session of parliament in modern times. They need it. We all need it. Theresa May certainly needs it.

But they and we also need to use the Brexit break to relearn the art of thinking straight and seizing the moment. The delay imposed by the EU on Wednesday night was not simply another familiar can-kicking exercise. Instead it may mark a pro-European turning point in the battle of Brexit. It is a moment not of enduring impasse but of real opportunity.

For the right, the aim was to bend the Tory party to their obsessional will. Instead they have wrecked their party

It is now nearly five months since May signed the EU-UK agreement on Brexit. Since then, the Conservative party’s rightwing nationalists have repeatedly tried to defeat the deal and to oust May. They have dominated the airwaves and won some famous victories along the way, but in the end, they have decisively lost the war.

For the right, the aim was to bend the Tory party to their obsessional will. They have failed to do that. Instead they may have wrecked their party.

Their aim too was to drive the UK out of the EU without a deal of the kind signed by May or any of the economic safeguards Labour and other opponents demand. That is not now going to happen. Wednesday’s agreement in Brussels makes that clear.

May and her deal are very badly battered. But they both survive. The rightwing nationalists react as if these are mere details. They talk as if they will still capture the party leadership over breakfast, rewrite the Brexit deal at lunchtime, abolish the Northern Ireland backstop at tea, and win a general election on a populist English nationalist and deregulatory platform in the evening. It is all a fantasy. They have lost. Their strategy is bust.

Q&A What is a 'flextension'? Show Hide Donald Tusk has proposed a 'flexible extension' to Article 50. Under the plan the UK’s nominal last day in the EU would likely be 10 April 2020, but Britain would be expected to leave well before then, as soon as a withdrawal agree had been ratified by the UK parliament.

Their bravado comes from a failure to understand the subtle change in mood that is happening in parliament and, to a degree, in the country. Even if they capture the leadership, which is not certain, their other triumphs are wholly imaginary. There is no majority for no deal among Tories, let alone among MPs more generally. On the contrary, the majority against the go-it-aloners is growing. Their bluff has been called, and they are too foolish and too dogmatic to realise it.

May’s decision to open talks with Labour was a watershed. Its importance was widely underestimated, because May herself is underestimated, albeit for understandable reasons. In fact, the approach to Labour marked May’s acceptance that the Tory-facing Brexit strategy she had pursued for 30 months had failed. The EU leaders see this, even if the British political class sometimes fails to. It was one of the key factors in another notably statesmanlike approach by the EU27 in Brussels and by Donald Tusk in particular.

Play Video 1:26 Theresa May defends Brexit delay in parliament - video

The object of her turn towards Labour is to find a more moderate Brexit that Labour and others could vote for. The possibility that she may succeed in doing this is too easily dismissed. These talks matter. May refuses to talk about erasing red lines, but it will have to happen – and it is. There is little doubt that the two big parties are fairly close on some form of customs union. The EU has explicitly said it would accept that.

Another key point is the need to stop any future hard-Brexiteer Tory leader shredding a bipartisan agreement. This issue is causing trouble in the talks. The simplest answer is for the Tories not to vote for such a leader in the first place, or for the public to defeat them in a general election. Another would be to give the deal treaty status with the EU. But why not also require any major change of policy to obtain the support of all four constituent parts of the United Kingdom before it can be implemented? No unionist could oppose that. Even the SNP would have to back it.

May’s turn to Labour has also opened up possibilities that she has always adamantly opposed. Much the most important is that, merely by talking, she has put a second referendum – in which Britain may vote to remain – into the realms of the possible for the first time since 2016.

Jeremy Corbyn does not want Britain to remain in the EU either. But his party overwhelmingly does, and the wider public mood on this issue is growing much more insistent. A second vote is the only realistic means of obtaining that outcome. And Labour has to keep its soft Brexit supporters and its anti-Brexit supporters on board. So any Tory-Labour agreement, therefore, has to address the second vote issue in some way.

There are many options on this. May’s mantra in her Commons statement today was that it can be raised as an amendment to any wider agreement. But a much stronger commitment would probably be a precondition for any Labour buy-in. Most Labour people want no Brexit rather than a softer one, so a second vote will be a decisive issue when the talks resume. The point to note is that the issue has now reached the top table.

May faces a dilemma over timetabling. The Tory party is desperate not to hold European elections on 23 May. It fears derision, desertion and defeat if it has to go through with them. May made her own opposition explicit in her Commons statement. Labour can probably be a bit more sanguine – the polls look less bad for them. Ministers therefore fear that Labour will spin the process out, so the government is forced to make further concessions.

What does a Brexit delay mean for politics, business, citizens and the EU? Read more

But 23 May could be an opportunity too. Most previous European elections have contained no debate at all about Europe, with the contests seen as an opportunity to kick the governing parties without suffering any penalty. This time could be very different, since there would be an intense argument about Europe, and the parties would face serious questions about their policies. That would be tough for both the Tories and Labour – and rightly so.

Today, the Berlin-based European Council on Foreign Relations published research showing British attitudes towards the European project have become more positive over the past decade. The UK is second only to Finland in feeling increasingly European. The 2016 referendum may have done more for our European consciousness than all the preceding years of neglect.

The one big plus of the past five months is that British voters have become much better informed about Europe and have thought more deeply about the issues. Last month’s march and parliamentary petitions showed the effect. So did the quality of those long House of Commons debates. There seems little doubt about the direction in which public opinion is gradually moving. These European elections could reflect it. It is time to be more confident again about Britain’s place in Europe. It’s time to go for it.

• Martin Kettle is a Guardian columnist