BBC presenter Justin Webb has said the media needs to look again at how it covers politics and the way it holds people to account in the wake of the vote to leave the European Union.

Webb, one of the BBC Radio 4 Today team, spoke out after Oscar-winning film producer Lord Puttnam criticised the BBC’s coverage of the European debate as constipated and effectively hamstrung by its own strict rules on impartiality.

Webb said some people who campaigned to remain in the EU had felt let down by the media’s coverage of the debate before the the historic poll result on 23 June. “A discussion about holding people to account, a discussion about impartiality in the modern era, is one I suspect the broadcasters would rather welcome, if only to sort out their own thinking,” the BBC’s former North America editor, wrote in the Radio Times.

“And it should not be a discussion left to newsrooms and editorial offices and university journalism departments: it really should matter to us all.

“One of the clearest messages during the referendum campaign was that audiences were hungry for real knowledge. People wanted to go beyond claim and counter-claim so that they could work out what was true.”

The aftermath of the vote has been marked by leading leave campaigners backtracking on claims made before the vote, such as the pledge by Vote Leave to spend £350m “sent to the EU every week” on the NHS.

Webb wrote: “Some of those on the losing side think they were let down. The Oscar-winning film producer Lord Puttnam is among those who wonder if impartiality rules torpedoed the search for truth: he accused the BBC in particular of providing ‘constipated’ coverage.

“The impartiality question is a reasonable one to raise – and it is one the BBC has grappled with on subjects such as climate change, where most scientists are on one side of the argument but some very feisty campaigners think they’re wrong. But the question has to be part of a wider debate.”

Puttnam, the former deputy chairman of Channel 4, said last week that media as a whole had failed to tackle the “Monty Pythonesque vision of Europe” which he said had been allowed to go unchallenged for the last 30 or 40 years.

Puttnam himself had been one of the driving forces to change the way the BBC reports on climate change.

The BBC’s former director general Greg Dyke echoed Puttnam’s thoughts at the launch last week of a report into the future of public service broadcasting. “I understand exactly why they ended up reporting it the way they did, because there’s people with stop watches and all the rest of it, but the result I thought … was a little bit dull to be honest,” said Dyke.

Webb said: “We tend to regard campaigning as promising policies or aspirations that can be tested against the facts of the real world. A combination of forensic interviewing and zealous fact-checking strips away the nonsense and allows the public to make a balanced choice.

“Seriously? In the modern world, this is not necessarily what happens. It is a truism to say we’re post-ideological: we don’t vote tribally for ‘the workers’ or ‘toffs’, based on a love for socialism or capitalism. It is equally cliched to talk of post-factual debate, where no one accepts the version of reality presented by anyone but their own side,” he wrote.

“Our real problem might be that we are entering, as the Americans seem to have entered, an era of identity politics where the politicians, the campaigners, are seeking by a process of nods and winks to let you know: ‘Hey, this is where you belong. Your people are here.’”