The elephant in the room. Everybody knew what Mark Carney meant when he paused halfway through his regular three-monthly update on the state of the economy: the implications of Brexit.

The governor of the Bank of England did not pull any punches. He warned of a potential run on the pound and of possible problems financing the UK’s whopping balance of payments deficit. He said the Bank expected growth to be materially lower and inflation to be notably higher. Voters trust the Bank of England. They sat up and took notice. The opinion polls started to move in favour of remain. When the history of the referendum campaign is written, Carney’s may be seen as the decisive intervention.

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In truth, there was more than one elephant in the room. Carney was right when he said there was a risk that the upheaval caused by Brexit could tip an already weakening economy into recession. But as elephants in the room go, this was the smaller, Indian version. The equivalent of the bigger, African elephant was the shocking state of the eurozone after the failure of the single currency experiment. This went unremarked by Carney, although it is relevant to the debate about Europe.

Why? Because, although Britain is likely to stay in the EU, Brexit will remain a live issue unless the eurozone can sort itself out. That means either admitting that the euro has been a terrible mistake, or going the whole hog and integrating further, with a single banking system, a Europe-wide treasury, and a democratically elected finance minister with the power to raise money in Germany and spend it in Greece. This is not going to happen any time soon, and perhaps never. Countries that joined the eurozone gave up a considerable amount of economic power when they adopted the euro, but they retained the right to raise their own taxes and make their own spending decisions.

Put ever-closer union in the same category as the Birmingham inner city ring road: it seemed a good idea at the time

Britain is not in the euro, for which we should all be thankful. But let’s be clear: staying in the EU means hitching the wagon to a currency zone unable to go forwards or backwards, and which will continue to struggle as a result.

The euro brought to fruition the idea of ever-closer union, a plan that dates back to the early 1950s. Lots of things considered good ideas back then are no longer considered quite so clever: system-built high-rise flats as the answer to slum housing; nuclear power to meet energy needs. Put ever-closer union in the same category as the Birmingham inner-city ring road: it seemed a good idea at the time.

Dan Atkinson and I spent the winter working on a book about the single currency commissioned in the wake of last summer’s Greek crisis. The brief was to look at what had gone wrong from a left-of-centre perspective; to explore the widespread disquiet about the way in which a country that voted in January 2015 for an end to austerity ended up seven months later being forced to accept even deeper cuts in wages and spending.

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The eurozone crisis is about more than Greece. It is about Italy, where the economy is barely any bigger now than it was when the single currency was introduced. And France, where unemployment is double the level of the UK or the US. And Finland, one of the most tech-savvy countries in Europe, where the economy is 7% smaller than it was before the start of the financial crisis. And even Germany, where an export boom and high corporate profits have been paid for by workers in the form of below-inflation pay increases.

Our investigations took us back to the last time Britain held a referendum on EU membership, when during the cabinet discussions Tony Benn warned that Britain was signing up for something that was undemocratic, deflationary and run in the interests of big business. “I can think of no body of men outside the Kremlin who have so much power without a shred of accountability for what they do,” Benn said.

Benn’s dystopian vision proved entirely accurate. When the architects of the new Europe looked to the future, they envisaged a new and better version of the United States of America. Europe would have all the good bits about the US – such as the economic dynamism, a large barrier-free market and a single currency – without any of the bad bits: the inequality, the high levels of incarceration, the poverty and the inadequate welfare safety net.

This dream lives on. Yanis Varoufakis, the deposed finance minister of Greece, thinks the eurozone could be recast along Keynesian lines, with the rich and strong countries obliged to provide financial help to the poor and weak. Good luck with getting Germany to agree to that.

Last year’s threat was Grexit. This year’s threat is Brexit. Next year’s threat will be something else

Economic policy has been relentlessly deflationary. The interests of bankers have been given a higher priority than workers’. Greece, Ireland, Portugal, Cyprus and Spain have been the laboratory mice in a continent-wide neoliberal experiment of a sort Tea Party Republicans in the US can only fantasise about.

Given the obscene level of long-term unemployment, the idea of Europe as the guardian of labour rights is laughable. The gap between the US and Europe has widened, not narrowed, since the launch of the single currency. Populist parties of both left and right are gaining in support. One left-of-centre argument against Brexit is that it would result in the breakup of the euro and by doing so set off a chain reaction that would lead to the next global crisis: a perfectly fair point. Those who fear that another recession and even higher levels of joblessness would threaten a return to the totalitarian politics of the 1930s are right to highlight the risks. Some on the left who want Brexit say that the time is not yet ripe.

The left-of-centre case for divorce is that Europe doesn’t work, is not remotely progressive and is heading for an existential crisis anyway. Last year’s threat was Grexit. This year’s threat is Brexit. Next year’s threat will be something else: Italy leaving the single currency, perhaps, or Marine Le Pen’s tilt for the French presidency.

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This presents an opportunity for those who believe that the way ahead still involves closer integration. Jean Monnet, the godfather of the EU, always said that ever-closer union would be forged through crises, which is what Brexit would undoubtedly trigger.

If the polls are right, Britain seems unready to trigger this act of creative destruction and it will be left to Varoufakis to do out of office what he could not do in power: prove a different Europe is possible.

A different Europe is needed, but it is stretching credibility to imagine that the Europe of Greece and the Transatlantic Trade and Investment Partnership can easily morph into America with the nice people in charge. The eurozone is economically moribund, persists with policies that have demonstrably failed, is indifferent to democracy, is run by and for a small, self-perpetuating elite, and is slowing dying. The wrong comparison is being made. This is not the US without the electric chair; it is the USSR without the gulag.

• Europe isn’t working, by Larry Elliott and Dan Atkinson, is published by Yale University press on 24 May