Former chancellor Ken Clarke writes in his memoirs that in British political history there is nothing so dead and forgotten as old budgets. That must be reassuring news for Philip Hammond. The current chancellor’s sticking plasters this week are best understood as a far from glorious 21st-century application of Harold Macmillan’s advice to Tory chancellors that “we must do something, or else the socialists will promise everything”.

This short-term approach lay behind the measures on stamp duty, universal credit, the NHS and the regions. But Wednesday’s modest handouts were dwarfed by the historic slowing of growth, and the consequently falling real wages and living standards over which Hammond presides. These will shape British life in the coming decade far more than any of this week’s budget measures. None of Britain’s political parties have the measure of this yet.

The budget certainly solved nothing in the macro-economy or in the centre-right’s deeply divided view of it. Hammond’s aims were modest: to avoid mistakes that unravel; to still be in his job at Christmas; and to prevent the premature self-detonation of Theresa May’s government. In that context the most important thing that happened on Wednesday was the approval that Hammond secured at the traditional post-budget meeting with Tory MPs. It was another day survived in the trenches.

The fact that a budget free of banana skins counts as an achievement is a mark of the underlying problems facing British party politics over the next decade. Hammond’s budget was about keeping a battered show on the road. It was managerial. It made only marginal adjustments. There were implicit long-term changes, of which the most notable was the tacit disavowal of the austerity economics embraced by George Osborne in 2010. But Hammond was not in a position, nor the politician, to tell a new story about the political economy of Britain to his divided party.

Few chancellors in memory have come into the budget period with their authority so under fire. That fire all came from the Tory party and the anti-European press. It was overwhelmingly about Brexit, not economic policy. The Brexiter right see Hammond as faithless to the cause. They have turned up the heat on him as a direct retort to challenges against Boris Johnson. Underlying all the recent challenges to Hammond has been the simple threat that if the pro-Europeans manage to take down Johnson, the anti-Europeans will take down Hammond in return.

The inability of the party to agree about both the public realm and Brexit is deeply intertwined. The failure of the banks, and the fateful choice to leave Europe that to some extent sprang from it, ensnares all attempts to plot a coherent path forward. Both failures tell voters that times will get worse not better. But voters remain uncertain about which party offers them the better protection. Both the Tories and Labour are struggling to cope with divisions on the issues, though the Tories’ problems are more visible.

The Tory right are in fanatical mood. They don’t care much who gets in their way or who they offend

This raises larger questions about the long-term adequacy of the existing parties and the electoral system to reflect the issues of the day, never mind resolve them. All large parties in Britain tend to be coalitions. The Tories still stretch from the liberal, patrician centre to the anti-tax, nationalistic right; Labour from the managerialist middle to the Leninist left. The existence of separate traditions within the same party is both a challenge and a reality, though these have only occasionally made either party unviable: the Tories over tariffs and Europe; Labour over nuclear weapons and union power. But the traditions need to respect one another’s legitimacy.

The Tory version of this problem is in plain view. Hammond has done something to bring his party back together briefly on economic and social policy. But he has nothing to say about the era of deteriorating public finances and the squeeze on incomes that now looms in the 2020s and perhaps beyond.

The Tory party’s internal coalition is also about to be tested to possible destruction by Brexit. It all adds force to the warning from their MP George Freeman that the Tories risk being defined as “a narrow party of nostalgia, hard Brexit, public sector austerity and lazy privilege”.

The Brussels summit in December is the next crossroads. May seemed to win space in the cabinet this week to negotiate the price for leaving the EU without being denounced as a traitor merely for trying.

But the right are in fanatical mood. They don’t care much who gets in their way or who they offend. They are thrilled that Angela Merkel is in trouble, fired with anti-Irish sentiment after Leo Varadkar’s warnings about the customs union, and their anger is mounting over the EU withdrawal bill’s struggles in the Commons, with more in the Lords still to come. If the British government ends up doing what Northern Ireland’s DUP wants, rather than what moderate swaths of the Tory party and its voters want, the political cost could be huge.

Labour, by contrast, has managed to keep its internal problems out of sight recently. That reflects well on the party’s right and left, which seem willing to manage their differences rather than go to war on them, though neither of them takes much joy in the arrangement. Jeremy Corbyn’s decision to focus on Brexit at prime minister’s questions this week, after ignoring the subject for so long, suggests he is in a more conciliatory mood towards the huge majority of MPs who want him to prioritise the issue. It is a good approach.

The divides are still there, potentially deep and destructive. Right now, though, Labour is moving steadily towards being the party of a very soft Brexit while at the same time not provoking internal conflict over economic policy by obsessing about specific pledges. The collective willingness to avoid showdowns may be more apparent than real, but it is sensible tactics. If Corbyn is to win a Commons majority he needs moderate remain voters in non-Labour seats to decide that the election of a Labour government is in the national interest.

In parliamentary systems, like it or not, either the parties must be broad churches or the governments must be coalitions. A choice must be made. You either make your deals and compromises before the election within a party, or afterwards in a coalition. Both main parties struggle with this reality and each is often in denial about it. But there is no third way for either of them.

• Martin Kettle is a Guardian columnist