Call me naive, but I always thought that if the opposition have a rare and clear chance to defeat the government, they would take it. Now we learn that sometimes, even when presented with an open goal, the opposition would rather kick the ball into touch.

The open goal in question is the arrival in the House of Commons next week of an amendment, passed by the Lords, that would keep Britain in the European Economic Area (EEA): in other words, a soft Brexit that would see the UK leave the EU but stay in the single market.

That matters, because single market membership is crucial to Britain’s economic health, and is surely the only way to deliver the “jobs-first Brexit” that Jeremy Corbyn has promised. Also, given that the referendum two years ago was so close, the EEA position would seem a fair reflection of the will of the people. (Arch-Brexiteer Daniel Hannan recently went further, hinting that the narrowness of the win might have triggered no more than a renegotiation of the UK’s EU membership rather than a departure – though he said that on Dutch TV, so perhaps he thought no one would notice.)

Tory rebels reckon they have enough MPs to win a vote on the EEA plan – but only if Labour backs it. Put simply, Labour has a clear chance both to defeat Theresa May and ensure a Brexit that limits the damage Britain is about to inflict on itself. It would be good for both country and party.

But now it’s clear that this is an opportunity Labour would rather pass up. the party released an alternative amendment, one that falls short of joining the EEA. Instead it talks of “full access to the internal market”.

At first, some remainers seized on that as a welcome shift by the party leadership towards the single market. But gradually a more sober view took hold. This was a way to derail a move that might actually have kept Britain in the single market – the EEA proposal – and was little more than a reheated version of Labour’s repeated pledge to deliver the “exact same benefits” of the single market and also to leave it, a position Labour’s own Barry Gardiner memorably described as “bollocks”.

So what explains Labour’s decision? Keir Starmer told the BBC’s Today programme it was merely a pragmatic acceptance of reality: Labour MPs would not unite around the EEA proposal, and this alternative clause is one everyone on the Labour benches can live with. But that’s worth unpacking.

For what exactly is the problem some Labour MPs have with the EEA? The answer is simple: immigrants. They worry that EEA membership would entail free movement of people, and several worry that their constituents, in leave-voting seats such as Darlington or Don Valley, won’t stomach that. Indeed a briefing note to MPs explaining the decision explicitly mentions freedom of movement as a problem with the EEA.

It would be interesting to hear Corbyn explain to some of his younger supporters – who are both pro-remain and relatively relaxed about migration – that he passed up the chance to soften Brexit because Labour wants to keep out immigrants.

MPs from those leave-supporting seats would say they have nothing against newcomers, but they have to bow to the reality of their constituents’ feelings. And that means scuppering the EEA proposal. But does it?

As it happens, Labour’s new amendment surfaced just as Gordon Brown made a valuable intervention. He said that rather than obsessing over the technicalities of Brexit, politicians needed to address the anxieties that had fuelled the leave vote. On immigration, that could be done, he argued, by a series of measures – from ensuring local people have a chance to apply for every job to registering migrants on arrival in the UK – that are all permitted under EU rules. Indeed, they include steps already implemented by that most European of EU members, Belgium.

Put simply, there is a solution here to Labour’s conundrum, as it seeks to keep both leave and remain voters onboard. It could back both the EEA referendum and Brown’s package of measures on immigration. That way, it would, to adapt a phrase coined long ago by Brown himself, be tough on Brexit and tough on the causes of Brexit. And in the process, it would have a chance to inflict defeat on an ailing, failing government. The question for Labour is: why doesn’t it take it?

• Jonathan Freedland is a Guardian staff columnist