Theresa May’s summit of party leaders has ended in acrimony after Jeremy Corbyn walked out of talks, citing the presence of Chuka Umunna of the Independent Group, who is not technically a party leader.

The meeting is an illustration of the strengths and the sharp limitations of May’s political approach. Her communications director, Robbie Gibb, tends to organise impromptu broadcast hits when the Prime Minister has done badly in the House of Commons and he did that effectively again today. May’s meeting with the six representatives made good footage for the six o’clock news and her statement – a boilerplate stop-me-if-you’ve-heard-this-one-before address – in front of Downing Street will lead the ten o’clock news. It will similarly be picked up by the newsbreaks on music radio. These are the important broadcast slots in terms of winning and losing elections.

It worked even better because Corbyn walked out of the talks. It advantages literally every political force not called “the Labour party” for Labour to be the sullen ones: the Conservatives want to paint them as wreckers rather than for the focus to be on their own splits, while the various pro-Remain parties and groupings all want to use the Remain/Leave divide to dismantle Labour’s electoral coalition to their advantage.

The problem is that, while the advantage is real for the pro-Remain forces in Parliament, regardless of how Brexit is resolved, it is illusory and temporary for May and the Conservatives. The only path to a Brexit deal passing the Commons is with a sizable infusion of Labour votes. That will only happen if the political toxicity to Labour of doing so is reduced.

That toxicity needs to be reduced in at least one, and ideally two places: it needs to be reduced among Labour activists, so that Labour MPs feel they can back a deal without being deselected; and ideally it will be lessened among Labour voters as a whole in order to get the leadership either on board or at least looking the other way.

It will do May no good at all that she won the headlines if, in nine days or at the end of a short transition period, the United Kingdom crashes out of the European Union without a deal. The beneficiaries then will of course be the pro-Remain parties, but also the party who has done the worst out of today, the party who thanks to our electoral system voters know full well they have to turn to if they really want to punish the Tories: Labour.