Fifty years ago, in an address to the nation, the US president Richard Nixon immortalised the phrase “silent majority”. In the febrile context of 1969, he was referring to the millions of law-abiding Americans who were not involved with, or sympathetic to, 1960s counter-culture, the protests against the Vietnam war and angry campus campaigning.

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Does the idea still have meaning? Social media, smartphones and the resulting digital cacophony have pretty much seen off the possibility of “silent” anything. But noise is not the same as democracy.

All the same there is, I think, something approaching a majority in this country that – however it votes in first-past-the-post elections – does not feel truly represented by either the nativist Conservative party or Corbynite Labour. And this group does not fare well in a political culture that favours polarisation and amplifies shrillness.

On Brexit, they may not be remainers in the sense that they feel a passionate allegiance to Brussels or the treaty of Rome. But they have observed the following, with growing anxiety: that the nation decided in June 2016 to follow the path to departure from the European Union; that the voters were lied to and electoral law broken in the preceding campaign; and that, after nearly three years, the path has led us all to the edge of a cliff. They wonder if it really would be “treacherous”, “anti-democratic” or a violation of the “will of the people” to pause at this political Beachy Head and – at the very least – reflect upon our next steps before we head over the precipice. That’s the great thing about cliffs. You can pull back from oblivion as often as you like. You can jump off only once.

So, with less than six weeks to go until the official Brexit date, this group feels not that it has “taken back control” but precisely the opposite. Polling by YouGov shows there is a clear majority against leaving the EU when voters are asked about specific forms of departure – “no deal” or Theresa May’s withdrawal agreement, for instance. Who speaks for these people?

There is an alternative path, and it is becoming clearer by the day. Taking it will not be easy, but this is no time to prioritise convenience or simplicity (the great sin of the populist).

First, the UK must seek an extension of article 50 from our 27 EU partners: the Commons will have the opportunity to lay the ground for such a request when it votes (again) on Yvette Cooper’s amendment on 27 February.

But that in itself will not be enough: the EU will not grant any such postponement unless we persuade its members that we are seeking something more substantial than a few more months of can-kicking. There has to be some sort of plan.

And that plan has to involve a final-say referendum. How many permutations of impossible do we have to watch the Commons consider before we all accept that there is no distinct parliamentary solution to this conundrum? The only proposal for action that MPs have approved – Graham Brady’s amendment on 29 January – was successful precisely because it was meaningless. Its demand for “alternative arrangements” to the Irish border backstop was simply another way of saying: we agree that we don’t have an answer.

Which is why, having voted for the Cooper amendment, MPs should also back the plan put forward by the Labour MPs Phil Wilson and Peter Kyle, and backed by several Tory remainers. This would give the Commons the chance to back a deal that would then be put to the electorate in a referendum. It combines the best of parliamentary responsibility and plebiscitary democracy.

Play Video 1:24 Corbyn calls on May to bring forward 'coherent' Brexit plan – video

There was a time when I believed a general election might break the impasse: but no longer. Jeremy Corbyn has made it overwhelmingly clear that he has no interest whatsoever in a people’s vote; indeed, that he actively favours Brexit, and the sooner the better. When he says the option of a public vote remains “on the table”, one thinks of all those copies of Stephen Hawking’s A Brief History of Time that were put on display, untouched, in middle-class households. If you believe Labour under Corbyn is ever going to give collective, official backing to a fresh referendum, you really haven’t been paying attention.

May, for her part, has become so fixated by what she sees as her historic duty to deliver Brexit that she would rather lose a general election than oversee a second vote that might thwart our EU departure. Many Conservative prime ministers, after all, have been kicked out of office by the voters: Edward Heath, John Major, even Winston Churchill. But if May were in No 10 at the time of a public vote to block Brexit she would always be damned in the folklore of the right as the prime minister who betrayed the dream of national independence. For a lifelong Tory such as May, this is an intolerable prospect.

Which is why I do not believe most cabinet ministers when they insist the “last thing” they want is a general election. For a start, they believe, with quietly growing confidence, that they can beat Corbyn unambiguously this time. And – remarkably – they’d rather take that chance than risk the wrath of the far right by acceding to a new referendum. Which, when you think about it, is seriously pathetic.

Yet, in these urgent and decisive times, another head-to-head between May and the Labour leader would simply not offer the choice that the nation needs. Perhaps uniquely in the history of modern general elections, it would be a sideshow. So – given this tragic abdication of responsibility by the two main party leaders – it is for members of parliament to shrug off the chains of tribalism, the discipline of the party whip and the growing threat of constituency deselection. If ever there was a time to think, without fear, of the public interest, and to set aside partisanship, this is it. We inhabit a digitised culture of political cantonisation, reflexive hostility and snarling defensiveness. Which is to say that what is needed is, in the truest sense of the word, counter-cultural.

There is still a chance for MPs to prove themselves worthy of the public trust, and a way of giving voters – including those who opted for leave in 2016 – a final chance to say whether this is really what they had in mind. A few steps away from the cliff’s edge, we still have time to think again. There is no direction to history, no implacable force of providence driving us towards an act of irrevocable, collective self-harm.

What happens now will happen because of human agency, or its absence. The parliamentary technicalities may be complex but the choice could not be simpler: what do we actually want?

• Matthew d’Ancona is a Guardian columnist