Tiny committees can have big impacts. Hardly anyone knows who is on Momentum's National Coordinating Group – but it is the cog that turns Momentum, and Momentum is the cog that turns the Labour conference. Its decision on Saturday not to take a position on Brexit ahead of this month's party conference could prove pivotal in the formation of Labour's final policy. Neutrality might be a step forward from last year, when it instructed delegates to actively block Brexit from being debated, but it will disappoint the thousands of Momentum members still petitioning for a proper ballot.

Below the surface of the headline politics, something else is going on. Momentum's ultra-loyalist wing – those who view it as a vehicle to support the Labour leadership line on any given question, and rationalise it to the party's base – mobilised to block a ballot of members because they believe that an anti-Brexit stance is not in the left's strategic interests. Yet in fact, they are boosting the prospects of the old Labour establishment and giving the party's right a route to recovery.

For the New Labour establishment, the game really ought to be up. Like social democratic parties across Europe, most of whom are now facing electoral apocalypse, Labour's old mainstream mortgaged itself to the political consensus of the era – privatisation, deregulation and austerity. Both its policies and its political culture are profoundly unpopular. The continuation of Thatcherism and the aloof, professionalised political model disintegrated communities and created the conditions for Brexit in the first place.

As it has always done in times of crisis, the Labour right is clinging to the independence and durability of the parliamentary party as the left-wing tide rises. But as mandatory reselection returns to the agenda, it is entirely plausible that the composition of Labour's MPs will change. In any case, the left will only ultimately lose control if it loses the base of the party – and the Labour right can only regrow and prosper if it can drive a wedge between the left leadership and its supporters, and push a narrative that puts it on the right side of history from members' perspective.

Brexit is the issue that could save Labour's centrists – if the left abandons the issue. As many as 78 per cent of Labour members, when polled, say they want a public vote on the terms of Brexit with an option to stay in the EU. As exit day gets closer, any hope of Labour delivering a Brexit on its terms gets more and more remote. Brexit began as a Tory project, and, if it happens in anything more than name, it will mean be a bonfire of regulations and rights and an attack on the prosperity of working-class people. Even if Labour comes to power tomorrow, there will be no bespoke Lexit.

At a grassroots level, the debate is moving fast. By the time Labour conference begins, hundreds of constituency parties will have debated anti-Brexit motions in a surge led by a growing network of left-wing and Momentum activists. But even as the realities of Brexit confront us, the aim of Labour's left pillars is to doggedly avoid the subject and even to triangulate towards the sentiments that motivated the Leave vote. At a higher level, the Labour leadership's position has changed little since June 2016.

It is absurd that the Labour right could present themselves as the guardians of open borders. Tony Blair, his heirs and his allies have a risible record of pandering to tabloid-level anti-migrant prejudice, and valorising the “concerns” of the “real” (read: white, middle-aged men with regional accents) working class. But it is nevertheless true that the end of European free movement – which is not an inevitable consequence of Brexit – would constitute the biggest expansion of border controls in recent British history. This was a policy endorsed by the 2017 Labour manifesto under Jeremy Corbyn.

For as long as the Labour left's aim is to avoid a genuine democratic debate on Brexit, and to awkwardly triangulate rightwards on the subject, Labour's centrists will be able to frame themselves as the guardians of cosmopolitan, pro-immigration politics. Even more important than the positioning is the momentum – and who is proved right by events. A generation of young Labour supporters, the vast majority of whom are anti-Brexit and pro-immigration, are watching.

We've been here before. In the 1981 deputy leadership election, Tony Benn won 81.1 per cent of the membership vote. The idea that the left might spend decades in the wilderness would have seemed unthinkable. And yet in 1988, Benn secured just 19.6 per cent of members' support against Neil Kinnock in the leadership election. The same grassroots members who were part of the Bennite surge turned on the left. Corbynism might seem unassailable in 2018, but that position is always fragile.

Tory Brexit is the only Brexit on offer. It will be a disaster for working-class people and the communities Labour is supposed to represent. There is no mandate for it, and the people must be given the final say on the matter. This ought to be common sense. Most Labour members already think it. And if the institutions and leadership of the left don't occupy that space, somebody else will – and New Labour could yet be reborn.