Even now, after all that’s happened over the past few days and with everything to come, Labour politicians and their aides cling to one of two excuses for their position on Brexit. The first will come most often from an MP for some kicked-about northern seat. “I voted remain, of course,” they generally begin, “but my constituents wanted Brexit.” And so, despite all misgivings, Brexit they shall have. Soft Brexit, naturally, as soft and as yielding as a goosedown pillow, because our clear-eyed, good-hearted representative looks at the tragedy at Honda and knows they want no more of that – but enough Brexit, they hope, to satisfy the voters’ appetite.

The second excuse usually comes from those closer to Jeremy Corbyn, by employment or inclination. For them, Brexit is something to be endured for the greater good of enabling Labour to kick out the Tories. So nail those six tests to the door even if they are, to use a technical term, “bollocks”; offer to help Theresa May with her deal; strike a tone both constructive and ambiguous.

In this way does a party that is overwhelmingly remain, from its voters to its members to its MPs to its frontbench, end up as the midwife to leave. True, there is a smaller number at each level who truly believe in a leftwing Brexit, or Lexit, but easily Labour’s biggest motive is a desire not to upset the electorate.

And I can see the logic. This will not be yet another column fantasising about how Labour is run by a cabal of revolutionary grandads all huddled together on some Kremlin-sponsored allotment to plot the downfall of capitalism. It plainly isn’t, although I would pay good money to see that film. Nevertheless, what may seem sensible tactics adds up to dangerous strategy. In its focus on the immediate demands of holding together a fragile political coalition – only heightened by the walkout last week of Chuka Umunna et al – it ignores this moment’s historic significance. It is all trees and no wood.

The jobs lost to Brexit and the havoc it is already wreaking in government are staples of the news, but only in the past week has there been serious talk about how it will reconfigure politics. Yet one of the greatest risks is that Labour will chuck away its position as the most interesting venture in mainstream European politics, merely to tie itself to the back bumper of a hard-right juggernaut.

Brexit was always a project driven by the right to enrich the right. That goes for its most fervent enthusiasts, Thatcherite throwbacks such as her former chancellor Nigel Lawson and Patrick Minford, the economist who in the run-up to the referendum blithely forecast that Brexit would “eliminate manufacturing … But this shouldn’t scare us. Britain is good at putting on a suit and selling to other nations.” It applies to the prizes the right seeks from leaving, such as scrapping paid holidays and other workers’ rights, as reportedly plotted by Boris Johnson and Michael Gove. And just look at how it is already using this moment to change our notions of who gets to live here and on what terms. Does anyone think that home secretary Sajid Javid would make such a show of leaving a British teenager and her baby to rot in a refugee camp in a failed state were he not hoping to become the first prime minister of Brexitannia? This will be the country that feels no shame because it’s too busy being fuelled by hate.

‘Unless something major changes, the end of next month will launch the Rees-Mogg revolution.’ Photograph: Leon Neal/Getty Images

When I voted remain in 2016, it was not out of love for such sorry characters as Jean-Claude Juncker, but because I didn’t want the UK to be remade in the image of Nigel Farage. Well, I lost and it’s no consolation that the only thing I got wrong was the former public schoolboy: my eyes should have been on a double-breasted jacket and a monocle. Unless something major changes, the end of next month will launch the Rees-Mogg revolution, a reconfiguration of British society as drastic as that begun by Margaret Thatcher.

Why would the British left so blithely enable a Tory project that seeks to cripple it all over again? Should you need a reminder of how disastrous it is for Labour to enshrine Tory arguments as orthodoxy, then just think back to 2010-2015, when no opposition politician could begin a TV interview without disclosing whether they were now, or had ever been, a deficit denier. Now imagine that happening on cutting immigration, on trade giveaways, on slashing taxes.

There are of course the aforementioned Lexiters, who just know 29 March will bring the death of neoliberalism, even though the neoliberals will be in charge. Who claim that cutting immigration from the EU will allow more people to come in from the Commonwealth, although there’s nothing to stop that happening today if May’s government wanted it (spoiler: it doesn’t). Who have never quite grasped that the origins of reactionary British politics lie not in Brussels, but in Britain. In its vaulting ambition and loose thinking, Lexitism most closely resembles gap-year self-indulgence – a flight of fantasy tried out by people secure in the knowledge they’ll never have to suffer the worst consequences.

I don’t believe Corbyn should have greeted the 2016 result by blowing a fat raspberry and pushing to remain, but as Brexit fails to get through parliament, he should stop pushing for a compromise deal. That would only allow the pinstriped mob to argue that we’re still taking EU rules but getting none of the voting rights. Instead, Labour should get behind a second referendum. Leave voters would not punish Labour at the next election anywhere near as badly as its remain base, according to polling from the TSSA transport workers’ union that has been presented to John McDonnell and others in the past three weeks. Just 36% of Labour leave voters rank Brexit in the top three topics they care about. For Labour remainers, that shoots up to 60%. The TSSA briefing notes: “If [Labour] fails to oppose Brexit … there is every indication that it will be far more damaging to the party’s electoral fortunes than the Iraq war.” Its Scottish MPs would face wipeout, while in London there would be heavy losses.

The threat that Brexit poses to the British left is aptly summed up by an essay published 40 years ago. In The Great Moving Right Show, the late Stuart Hall laid out the scale of the challenge he believed the left faced from Thatcher – months before she even moved into No 10, years before she began her scorched-earth economics. But Hall saw it all coming: the populism of Thatcher, the way she would target schools and policing. And he saw how Thatcherism would win mass support: “Its success and effectivity does not lie in its capacity to dupe unsuspecting folk but in the way it addresses real problems, real and lived experiences, real contradictions – and yet is able to represent them within a logic of discourse which pulls them systematically into line with policies and class strategies of the right.”

Just like Thatcher, the Brexiters are poised to define the present, rewrite the past and then shape the future. It would be folly for Labour to aid them.

• Aditya Chakrabortty is a Guardian columnist