Donald Trump’s outburst may have done Theresa May a fleeting favour. Had the grand Shrek not delivered every imaginable insult (short of impugning St Gareth of Southgate) to his host country yesterday, the story in the spotlight this weekend would have been on the growing disquiet around May’s handling of the Chequers agreement on Brexit, and the darkening mood that has descended on her own benches. As it turned out, May rode out the turbulence. But with the awkward visitor gone, the stony road to Brexit – “a tough deal”, as the US president observed – resumes.

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What started a mere week ago as applause for the prime minister in facing down her most troublesome ministerial insurgents has slipped into acute agitation. It turns in part on the convoluted deal itself – but also on a fresh bout of panic about her ability to lead when the pressure is on.

The departures of Boris Johnson and David Davis disconcerted Brexiteers – but did not unleash rebellion. Thursday’s white paper was another matter. Its use of the term “association agreement” (not used previously) was a red rag to many bulls. Given that the last one the European Union signed was with Ukraine, it hardly takes a marketing genius to see the problem.

In this endgame of competing impossibilisms – hard Brexit versus a byzantine arrangement of near-customs-union “associations”, segmented agreements on goods and services, and somewhat indeterminate reassurance for the City on how its practices will be affected – the prime minister’s nightmare is that both enemy camps conclude they don’t want whatever she is offering.

This is the Zero Dark Thirty moment at which a serious move to oust May becomes probable – unless she can take back control of her disputatious party. May is not quite at that point – but perilously close. As one recently departed senior figure put it, there is no such thing as summer relief “because Graham Brady’s letter box is open over the recess”. Brady is the chair of the backbench MPs’ committee to which no-confidence votes would be submitted.

Facebook Twitter Pinterest US special forces on the hunt for Osama bin Laden in the film Zero Dark Thirty. Photograph: Universal Pictures/Sportsphoto/Allstar

A week of post-Trump recovery and a trade bill in parliament should see May through to the recess on 24 July. But the autumn is perilous – as Conservatives have to face their bewildered (and dwindling) grassroots at the party’s autumn conference. If May has no progress to report with the EU by then, the skies will darken.

“Taking back control” is a phrase being pumped out to brand the May proposals as the fruition of Brexit aspirations – a mistake, since it serves only to remind leavers that the present offer is a watered-down version of the freedoms they seek. Also, as Jacob Rees-Mogg notes, Trump is fundamentally right about the constraint on a future UK-US trade deal. The white paper makes clear that a common rule book on goods would make any deal with other countries subject to EU approval.

Some criticisms of May are less reasonable than others. Any prime minister having to enact Brexit would displease a vast number in their party and the country. A semi-demi-Brexit of the kind May has embraced is a rational response to that – but pleases no one. Style matters a great deal. As one recent cabinet minister (not a Brexiteer) puts it: this means she is “not selling anything to anyone”. Here is the most damaging criticism of the May method – it relies too readily on doing nothing much, except for drafting documents to send to Brussels and holding meetings. “I always have the feeling,” says one of the newly elevated ministers about encounters with her, “that she is backing out of the room from the moment we start.”

Truly, it would take the world’s most accomplished snake-oil vendor to retail a version of Brexit that is customs-union-compliant to solve the Irish border and trade impediments, yet reboots Blighty as a buccaneering free-trade power doing deals with an unpredictable US president. Still, May’s traditional weakness, reticence, has returned to haunt her, and Downing Street’s media strategy has been so risk-averse it doesn’t look like it trusts its own message. Sending Michael Gove on to the BBC’s Andrew Marr show last Sunday, with the rest of her top team confined to barracks, only attracts more incoming fire.

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Chequers was a bold play by the prime minister to get rid of her most toxic internal foes – and it demonstrates her capacity for ruthlessness. But her internal logic is not good enough: she needs to tell us why she wants her new deal and what the past two years have taught her. And she badly needs to level with leavers about the limits of what she can deliver for them – or unleash on herself another round of death by a thousand cuts.

May does have some capital to spend – having tied in Dominic Raab, an ambitious and bright leaver, as her Brexit secretary – an un-momentous role, but symbolically useful. Gove’s intellectual heft and resilience are respected by Tory leavers and remainers, even if he is evasive on precisely what he wants.

A wider hazard is that May’s hesitations spawn a mood of half-heartedness in her top team. As foreign secretary, Jeremy Hunt is a welcome replacement for Boris Johnson, who had evident contempt for the role. Hunt’s flaw is sounding perpetually apologetic about his views. That saps belief in compromise solutions, and emboldens those seeking even more impossibilist ones. “We must bring the country together behind a clear vision,” said the chancellor, Philip Hammond, in the Financial Times, gamely trying to outline a plan for financial services and regulatory triage. But May’s voice needs to cut through the din too, because in such febrile times politics is about filling the space of leadership – before someone else does.

• Anne McElvoy is senior editor at the Economist