The hills were alive with the sound of humiliation. At Salzburg, Theresa May was hoping she’d hear if not sweet music, then at least enough warm words from European leaders to allow her to say her Chequers plan still lived. Instead, they told her it “would not work”. Stung by that, she took to the Downing Street podium on Friday to deliver an icy death stare in Brussels’ direction – and to demand that the EU treat Britain, and her, with “respect”. There was even a hint that talks could break down, if the EU did not explain its objections and come up with counter-proposals.

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We are, then, approaching that point every negotiator most fears: when one side’s maximum falls irretrievably short of the other’s minimum. Lacking a Commons majority, constrained by her dependence on the Democratic Unionists and the veto-power of the Moggites, there is only so far May can go – and that is not far enough for Donald Tusk and the EU27. Chequers would have Britain sharing a common rulebook with the EU on goods, but not services. That’s simultaneously too Europhile for the Moggites and unacceptable to the EU, because it means cherrypicking the best bits of the single market. (The idea that this objection requires explanation, as May claimed in her monologue of cold fury, is absurd on its face: the EU has literally spent years explaining that the four freedoms are indivisible, that no member state can just pick and choose the one or two that suit them.) And all this before you even get to the impossible geometry of the Irish border question, where every permutation is unacceptable to someone.

The usual pattern with Euro-stalemates is for gridlock to endure until the 59th minute of the 11th hour, when the leaders finally emerge from an all-night summit clutching a bleary-eyed compromise. But that is becoming increasingly hard to imagine. Britain is asking for something that might just be impossible – which means the prospect of a no-deal Brexit is growing. Tellingly, May repeated her line that “No deal is better than a bad deal”.

That encourages the hardcore Brexiteers, obviously, but it gives some cheer to the hardcore anti-Brexiteers too. Their demand for a people’s vote rests on several premises that are looking ever more robust, buttressed by signs that we’re about to crash out. One is that the politicians have failed to deliver Brexit and it’s time for the people to take back control. Another is that a core promise of the leavers was that Brexit would entail a smooth departure: they needed us more than we needed them, they wanted to sell us their BMWs and prosecco, a Brexit trade deal would be the “easiest in human history”. Now it’s clear that all that amounted to a false prospectus, the people deserve to have another say.

For the avoidance of doubt: I buy these arguments, along with the rest of the case for a people’s vote. I agree that 2016 merely settled the principle of Brexit, not what it would mean in practice. I agree that if every trade union is required, under Conservative-passed employment law, to ballot its members first on strike action and then again on the final, negotiated deal, then the nation should at least have the same right on a matter of such gravity. I agree that departure from the EU requires “continuing consent”, that democracy should never shackle a nation to one decision, taken one time. I accept all that, but I have worries about a people’s vote all the same. Put my pessimism down to the dizzying events of the last two years, which have taught me to prepare for the worst.

Play Video 0:47 Brexit: 'No deal is better than a bad deal' says Theresa May - video

First, remainers tend to assume that the current chaos and craziness – including the humiliation of May at Salzburg – will put British voters off Brexit. But it could just as easily put voters off the EU. Plenty will like May’s tough talk at No 10, thinking it is those stubborn Europeans who are at fault, rather than our own leaders for demanding cake, unicorns and pipedreams. Who wants to be a member of a club that behaves like this anyway, they will cry. The current impasse could boost support for no deal, as more voters conclude that we should be “out and done with it”. According to YouGov polling conducted for the People’s Vote campaign, remain currently beats no deal by 57% to 43%: it’s a lead, but hardly insuperable. Given all the dire warnings about crashing out – no food on the shelves, no planes in the sky – that 43% favour it should give anti-Brexiteers serious pause.

Which gets to the heart of my angst. Let’s say the drive for another vote is successful. What would be on the ballot? What if it’s Justine Greening’s call for a three-way choice between whatever deal is patched together, no deal and remain: how confident could we be that remain would win that? It is at least possible that no deal would be a lot of voters’ second preference and emerge the winner.

This week’s admirably clear report, The Roadmap to a People’s Vote by the former mandarin and article 50 author John Kerr, seeks to reassure nailbiters like me by saying that any choice is likely to be binary. Could that be a Noel Edmonds referendum, offering the public a straight choice: the deal or no deal? No, insists Kerr: it would be “deeply undemocratic” to exclude the option of staying in the EU when that option commands such strong public support. I lack his confidence: undemocratic it might be, but I can still see it happening. I can imagine the arguments, as leavers insist that the question of whether to leave was settled in 2016: now all that remains to be decided is how.

But let’s say my fears are misplaced and that the eventual choice is a straight fight between no deal and remain. Again, I’d be nervous. Given the populist currents swirling in this country and beyond, given the appeal of a Brexit message that would say “What part of leave did you not understand?”, I can see no deal winning that one too – gaining legitimacy for what would be an act of national self-mutilation.

None of these is an argument not to go ahead with a people’s vote. Without one, this journey could well end in disaster anyway. A referendum is our best shot at averting calamity. My warning is against seeing it as the magic wand that will wave all our troubles away. A people’s vote won’t be the end of the war against Brexit – it will be the start of the biggest battle of all.

• Jonathan Freedland is a Guardian columnist