It’s been even more humiliating second time round. The United Kingdom has again been reduced to asking the European Union for an extension to the Article 50 process. Once was bad enough but twice marks a profound failure of government and Parliament. It has left the EU deciding the country’s future.

In Westminster, there is no sign of a resolution to the Brexit impasse. Cross-party talks between Labour and the Tories continue. Sources close to those talks feel that a common position is unlikely to be found. A deal between the two parties would require that Theresa May and Jeremy Corbyn both be prepared to split their parties. This seems unlikely.

When May met with the executive of the 1922 committee of Tory backbenchers on Monday, she went out of her way to reassure them that she knew how difficult a customs union would be for the party and stressed that she was looking at a customs arrangement instead. According to those present, May did not sound like a leader who was about to try to persuade her party to swallow a customs union.

Progress is also being hampered because MPs believe there’s now little chance of no deal. They think that Yvette Cooper’s law means the government can’t make no deal its objective and that the EU will never force it on the UK. Those MPs who don’t want no deal — 400 of them voted against it in the indicative votes — calculate that this makes it safe for them to carry on holding out for their perfect Brexit outcome rather than compromising. As one cabinet minister points out, if those who want a second referendum and those who are happy to leave without a deal won’t back any other option, it becomes very difficult to get a majority.

It is very hard to see how this log-jam can be broken in this parliament. The arithmetic is against any deal going through. An extension won’t, on its own, change the numbers. But there is one route being talked about with increasing frequency in Tory circles: a change of leader and a general election.

The theory goes that if the EU grants the UK an extension to the end of the year or for another 12 months, then May would go. To date, senior figures in the party have not wanted to force her out for fear of creating even more chaos. But I understand that this attitude would change if the UK were given a relatively long extension, because this would mean there was time to replace May without plunging the Brexit process into greater confusion.

An extension of more than a few months would give the Tories time to elect a new leader, and for the new leader to show where they wanted to take the country. They could then fight an election on their domestic policy platform and the end of free movement. If they won a majority, they would be in a better position to deal with both parliament and the EU — and to get a withdrawal agreement through.

This plan is high risk. There’s every chance that the public could use an early general election to give the Tories an almighty kicking; a change of leader might not be enough to save the Tories from the voters’ wrath at how badly the Brexit process has been handled. Even if the Tories won a majority, it would have to be their biggest in more than 30 years to guarantee getting a deal through: 34 Tory MPs voted against the withdrawal agreement at the end of last month.

Another danger for the Tories is that their leadership contest could be so acrimonious that it would be impossible for the party to come together again. To date, the putative candidates have been better at criticising each other than making a case for themselves.

Add to this that a leadership contest during an extension would be dominated by Brexit, the subject on which Tories are most likely to fall out with each other, and it is easy to see how a change of leader could leave the party even more divided than it is now.

If the Tories could elect a new leader and then come back together, the new prime minister would quickly have to craft a policy agenda. The last few years have seen painfully little action on the domestic front. The Christian Democrat approach the Tories took in the 2017 manifesto was discredited by the bad result. What can replace it? The Tories need clear, big ideas to take into any contest. They would need to show that they have an agenda for a fourth term in office that goes beyond Brexit.

In normal times, all of these risks — let alone the danger of Corbyn becoming prime minister — would mean that this strategy would never be attempted. But the country cannot carry on being stuck in Brexit limbo. Something needs to be done and if parliament won’t vote through the withdrawal agreement or anything else, then it is hard to see what options there are other than a general election.

There is no guarantee that an election would solve everything. Another hung parliament would most likely be another useless parliament as far as Brexit is concerned. Even a majority for Corbyn might not provide a definitive mandate for any Brexit position, given the Labour leadership’s desire to maintain ambiguity as to whether it wants a soft Brexit or a second referendum.

A general election would be dangerous for Brexit. A Tory majority under a leader who favoured a Canada-style free trade deal with the EU would lead to more Brexit. But any other outcome would lead to less Brexit, very possibly no Brexit at all. If Labour’s manifesto promised a referendum or if Corbyn had to commit to one to secure the confidence of the House of Commons, then the country would end up voting on a choice between Remain and a soft Brexit.

This is the reason why it would have been better for Brexit if the Commons had voted for the withdrawal agreement, guaranteeing that the UK would leave the EU, and then a general election had been fought on what kind of future relationship to seek with the EU. But there is next to no chance of that now. If Brexit is to happen, it must win again at the ballot box.