The votes in the 2016 EU referendum had not been cast, let alone counted, when some were already calling for the public to take to the polls for a second time. The health secretary, Jeremy Hunt, penned a column in the Telegraph to this effect just days after Brexit got the go-ahead. Labour’s David Lammy took the same view: “We need a second referendum at the very least,” he proudly declared, encouraging the British electorate to sign a petition that went on to receive more than 4m signatures. On Tuesday Tony Blair continued to echo this sentiment.

At first it was understandable, given that the leave campaign was characterised by foul play and a xenophobic undercurrent, and evidence suggested leaving the EU would be a disaster for Britain. But in the two years that have followed, many of those once wedded to the idea of remaining have taken time to reflect on how the vote came to pass. A vocal minority have failed to do so, preferring to repeatedly hound the Labour leadership for not demanding another referendum before the terms of Brexit materialise, and despite the British public, including many Labour voters, backing leave.

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There’s a simple lesson that the likes of Owen Smith are failing to learn: if you want attitudes to Europe to change in Britain, demanding Jeremy Corbyn commit to a second referendum was never, and is not now, the answer.

It has been Labour’s responsibility to point out that those who voted to leave the EU for a better future were sold a lie by a leave campaign that had no plan of action, and that ministers are gambling with the security and livelihoods of generations to come. Demanding a second referendum from the outset would have been a distraction from this task, giving credence to those Brexiteers who would say Labour opposes leaving the EU not because it’s proving harmful to the population but for dogmatic reasons, despite the “democratic will of the people”. The Tories are digging their own grave. Labour must continue to let them – the latest figures from YouGov suggest 50% of leave voters don’t think Brexit is going well.

Demanding a second vote from the offset would also have proved futile unless the attitudes that underpinned the Brexit vote were first challenged head on. If there had been another vote immediately, is it really credible that the public would have reacted in any other way than with outright fury? Much of the remain campaign in 2016 revolved around the economic impact of Brexit, but for many voters who had lived through longest decline in living standards in modern history, the status quo offered little comfort. To change attitudes on Europe therefore, progressives must also focus on taking on the social arguments harnessed by the right in the referendum.

Take, for instance, immigration – by all accounts a significant issue in the Brexit debate. It is only by challenging the anti-immigrant sentiment the right has whipped up that minds will be changed. This is a task that Labour has for too long neglected.

The British public’s outrage after the Windrush scandal is a case in point. Diane Abbott and David Lammy have helped shift the terms of debate in Britain around immigration – Theresa May’s flagship hostile environment policy has been disowned, Amber Rudd forced into a resignation. The Labour party should be focused on reminding the electorate that migrants – far from being a hindrance or drain on our society – are the very people we all rely upon to keep society functioning. And, consequently, this can slowly be built upon; the same is true for European nationals who for so long have been demonised by the right.

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Similarly, economic arguments must also be developed. A radical policy agenda that tackles inequality in Britain can prove to voters it’s our rigged system, not the EU, that is to blame for households struggling. There’s a reason many in the “centre” of British politics are so wedded to shouting about remaining – they’re out of ideas and unwilling to embrace Labour’s transformative, progressive policies.

If you want to see Britain take another look at Brexit, Labour’s path is obvious: highlight the mess the Tories are making, and convince people international cooperation is worth embracing. Don’t simply demand Corbyn wave an EU flag without even considering what a second referendum would look like.

Ideologues in the leave camp are determined to drag Britain out of Europe whatever the consequences. Labour must not mirror them by just shouting at voters that they’ve made a terrible mistake and seeming to be ideologically wedded to the EU come what may. The party must continue to prove that this Tory Brexit will be a disaster, and challenge the narratives that facilitated it. Once the final terms of May’s botched Brexit are presented, Labour can demand, if necessary and there is public appetite, that we have a say over any Brexit deal – whether in the guise of a referendum, a general election, or by some other means. The same old we-told-you so politics will only alienate swaths of the electorate, and ensure the Tories have no real opposition for their disastrous hard Brexit plans.