Here comes the prospect of redemption, a second chance for Labour to start over. A victory for Andy Burnham in the Makerfield byelection not only opens the door to No 10; a leadership contest also allows him and Wes Streeting to finally stretch their wings. Ideas currently firmly chained up in a Downing Street dungeon could be freed. Land value tax? Wealth tax? No more children in temporary accommodation? A national care service? Why not?

Not to be outdone, the government itself has unleashed a burst of activity, with Rachel Reeves’s summer of fun, as well as speeding up a deal with the EU and online protection for children. Expect renewed effort on nearly a million Neets (young people not in education, employment or training) with radical plans from Alan Milburn this week.

Some world-weary scepticism may greet offers from leadership contenders. Among 10 pledges Keir Starmer made on the hustings, the following were broken: the abolition of tuition fees; common ownership of mail, energy and water; an increase in income tax for the top 5%; and defending “free movement as we leave the EU”. (Key promises have been kept, such as workers’ rights and “no more illegal wars”.) Once inside that black front door, policies can prove hard to implement. Times change, Treasury coffers become empty, borrowing costs rise, Donald Trump starts an economy-killing war and heavyweight financial bullies put on the frighteners: stuff happens or nerves fail.

But this time something is very different. The single greatest pledge Burnham makes – and he has said it for years – will change British politics for ever, if he can win this byelection: early polling is very close.

Far more hangs on this result than (yet another) change of prime minister, or even the prospect of rescuing Labour from the threat of extinction. What could transform the future is Burnham’s strong commitment to electoral reform. Replacing the derelict first past the post system with proportional representation (PR) would accommodate our new five- or six-party landscape. But, above all, it would prevent ever again absolute dictatorship by one party with only a small minority of the vote. In 2024, Labour took 63% of seats on only 34% of the vote; that could not happen under PR, which apportions seats according to votes.

For years, the Electoral Reform Society has reported majority support for PR, but expect a new rise in interest at the prospect of Nigel Farage arriving at the Downing Street lectern on less than 30% of the vote. Burnham stands as the one to prevent it, the only truly popular Labour figure. Though Streeting has, in the past, sometimes sounded moderately in favour of PR, so far he has avoided the issue, despite it being the single change that can rescue British politics.

Burnham has spoken to me of his experience standing as Greater Manchester mayor, which has previously used a supplementary vote system, meaning voters get a first and second choice. “It completely changes the way you discuss politics with voters,” he says. When canvassing, if someone won’t be voting for him, he solicits their second choice, which opens a whole new conversation that he described last week as “less point-scoring, more problem-solving”. The most unpopular candidates can never win, obliging them to open their minds and avoid extremes.

A monumental constitutional reform such as PR would need the legitimacy of writing it into a manifesto, Burnham says. If he arrives in No 10, he should speedily summon a national commission as demanded last week by more than 60 Labour MPs. Let some wise head such as Sir John Curtice lead it to rapidly select a system: years of discussion and the previous Jenkins commission have left ready-made PR options.

Then call an election promptly: prime-ministerial honeymoons are becoming vanishingly short. Writing a new manifesto would earn Burnham personal authenticity and give him the authority to affirm his policies. Electoral reform is the most pressing by far, but he could confirm to bond markets that he will stick to Reeves’s borrowing rules, if not necessarily her precise tax plans. He could leave open the exact form of future relations with the EU, not ruling out a customs union, a single market or alignment through the European Economic Area.

Cleansing cash from politics would be a timely clarion call, as Farage has broken all trust in the guardrail between personal profit and political fundraising. He has normalised taking vast donations and earning unthinkable side-hustle sums from his political status. The Guardian revealed that in 2024 £5m came to him personally from Christopher Harborne, the Thai-based crypto billionaire who has bombarded Reform UK with cash. It doesn’t look good that Reform’s proposed “cryptoassets and digital finance” bill will cut crypto capital gains tax to 10%, force the Treasury to build a bitcoin reserve fund and allow taxes to be paid in crypto. It may not be illegal to promote your donors’ interests, in this case potentially boosting the use of cryptocurrency, but it stinks.

The public shock doesn’t feel deep enough. Farage follows the brash and greedy instincts of rightwing leaders from Donald Trump to Boris Johnson and Silvio Berlusconi. Remember Trump once said: “I could stand in the middle of Fifth Avenue and shoot somebody, and I wouldn’t lose any voters.” Either people think all politics is corrupt anyway or gold adds glamour to their favoured leader, or else they think any untoward behaviour is all a mainstream media lie.

Electoral reform and pressure-washing money out of politics is how to beckon back Liberal Democrat and Green defectors, ensuring their willingness to vote tactically again. They will know that a PR future would result in coalitions including them. They know the elemental divide is between the right bloc and the left bloc. They don’t need to think Burnham is the messiah, but he is the only likely route to rescuing Britain’s warped and corrupted politics.

• Polly Toynbee is a Guardian columnist