Theresa May doesn’t do spontaneous. Most of the time, to the delight and profit of the parliamentary sketchwriters’ guild, she is scripted and self-disciplined to a fault. She repeats her lines until the repetition is a form of torture. So how do we explain it if, just once, she strays momentarily off piste?

Don’t get this wrong. Off piste for May is an extremely relative concept. It is not the same as for Boris Johnson. Blink and, in May’s case, you miss it. Most of the time, the vicar’s daughter’s approach to her job seems to echo the words of Lucy Baldwin, wife of the Tory prime minister Stanley, who, when asked at Lord Curzon’s funeral in 1925 whether she was a believer, replied: “I am indeed, and I must tell you that every morning when we rise, we kneel together before God and commend our day to Him, praying that some good work may be done in it by us. It is not for ourselves that we are working, but for the country and for God’s sake. How else could we live?”

Nevertheless, there was no blinking or missing the brief spark of anger and combativeness this week, when May was challenged in the House of Commons by Labour’s Heidi Alexander, who accused her of “running scared” of her party’s extreme right over Brexit. May got to her feet, threw her briefing notes on the desk, said “whoops” as she gathered up her straying papers, and glared at the Labour MP. Alexander, May insisted with unusual emphasis, “could not be more wrong”.

In cabinet there are sharp divisions. May and Hammond are on the far better, cautious side

For May to react in that way is tremendously out of character. So the more one reflects on why she had that brief and uncharacteristic rush of blood in the Commons, the more it feels as if Alexander touched a nerve.

If that’s right, then the thing that matters about the exchange is surely this. May is weakened and under pressure. She had a rotten party conference, and Johnson and the press are incorrigible troublemakers. But the big thing May thinks she is doing as prime minister, as the moment of truth on Brexit gets closer, is to stand up against the dogmatic Brexiteers while trying to hold the Tory party together.

That moment got a little nearer on Thursday when it became clear that the European commission and the UK government have failed to reach agreement on the first tranche of Brexit talks. That outcome, confirmed in Brussels by Michel Barnier and David Davis, is no surprise to anyone involved. Both sides have been saying for ages that December is the crunch month on the issues – citizens’ rights, money, and Ireland – that the EU has prioritised. Even that deadline may be pushed back.

This reality is what lies behind the exaggerated and in some respects confected argument that is supposed to have taken place this week between May and the chancellor, Philip Hammond, over the extent to which Britain should start to prepare for a failed article 50 negotiation and a “cliff-edge” breakdown between the two sides – the outcome that the dogmatists in the Tory party crave so irresponsibly.

Facebook Twitter Pinterest Theresa May and Philip Hammond. Photograph: Hannah Mckay/Reuters

The Daily Mail attempted to portray the chancellor as a traitor in a daggers-drawn battle with the true-believing Brexiteer prime minister. In fact, May and Hammond are far closer on this issue than is characterised.

It is hardly a surprise – let alone wrong – that the chancellor is risk-averse on this issue, especially a month before his budget and with the finances looking very tight. Crucially, though, the two are on the same side over the necessity for the UK to require transitional arrangements between Brexit in March 2019 and the operation of formal new EU-UK relationships – the issue on which they carried the cabinet in September. The real knife fight is between May and the majority of the cabinet on the one side, and the fanatics on the other who actively desire no special new relationship with the EU.

If that’s right, then pro-Europeans of all kinds, Tory or not, have a champion in this fight, and that champion’s name is Theresa May.

The issues at stake here for pro-Europeans are multiple. One is to prevent the “cliffites” getting their way, which is to drive the UK into the arms of Trump’s America and move closer to the bonfire of the regulations that is their other ultimate goal.

A second is to promote the softest possible Brexit, in terms of UK relations and arrangements with the EU single market and customs union, that best preserve the jobs, living standards and economic prospects of British people – while also placing the right priority on keeping Ireland at peace. And a third is that supporting May on this issue is the best way – not an ideal way, but the best there is at the moment – of keeping the EU remain ball in play for the future.

There are other reasons for still taking May’s form of Toryism seriously and hoping that it eventually gains the upper hand over the deregulatory, small state right. One of these was on show this week, in the shape of the bill to control energy prices that May foreshadowed at the party conference.

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Another, in Washington, was the International Monetary Fund’s embrace of the viability of increased income taxes on high earners in advanced economies such as Britain. That’s a call that some Tories, including the former Cameron-era minister Oliver Letwin in his new book Hearts and Minds, have embraced too. While May remains prime minister, it does, at least, stand some chance of being considered.

But the massive importance of Brexit is the clincher. Within the cabinet, the party and the country there are fundamental divisions about what Brexit means. You don’t have to admire or support May to recognise that she and Hammond are on the far better, more cautious, side of those Conservative divisions.

On almost every issue that matters in the modern world, May’s broad approach towards the UK’s post-Brexit relationship with Europe remains more desirable and responsible – in spite of all its many faults and huge limitations – than the approach her opponents would put in its place if they could.

None of this makes May a great prime minister, a successful leader or a big thinker. Heidi Alexander was right to challenge her. But May’s response was revealing. Right now, and as the big calls on Brexit draw closer for the government, May remains, however improbably, the best hope the Tories have got – and even, while this government remains, of others too.

• Martin Kettle is a Guardian columnist