The BBC is on a war footing. As soon as the referendum on Europe was confirmed, an email pinged into journalists’ in-boxes from James Harding, the director of news, outlining the scale of the challenge in what he called “the heated months to come” – and telling them that they would be given training to get it right. Harding trumpeted the need for impartiality and promised to withstand political pressure. The campaigns have already been to New Broadcasting House to talk about how they will operate and, in the case of those in the Leave contingent, to urge the BBC to refer to “the EU” rather than “Europe”, to make it clearer what they don’t like and what they do. Such is the ability of politicians to find bias even in innocuous things.

The BBC deployed its multitude of platforms to launch its coverage, scheduling news specials on Radio 4 and BBC2 alongside the live streams. The only blip was when it was scooped by the defector Robert Peston on ITV. He got Boris Johnson’s decision to opt for Leave ahead of the papers and 16 hours before his BBC rivals.

Otherwise, it was a confident start. But the truth is that Europe has long been bothersome for the corporation. It is the agreed line that in the past, “The BBC was slow to give appropriate prominence to the growing weight of opinion opposing UK membership of the EU,” as one of the BBC Trust’s impartiality reviews noted. This is a fault seemingly located in the early 2000s, when I was head of television news. It is true that we were slow to spot the rise of Ukip but not, I think, Tory Euroscepticism: there was a period of pre-eminence on the airwaves for the Maastricht rebels and the John Redwood/Michael Portillo wing of the party. But it may have been a failing caused by the sense that the harder-line Eurosceptics did appear to be eccentric and obsessive.

John Major wasn’t alone in hearing the flapping of white coats. I once took a senior executive for lunch with a Tory Europe-basher whose monomaniacal tirade made my companion roll his eyes in despair every time the MP looked away. It was a similar experience when colleagues were buttonholed by Lord Pearson of Rannoch, an early supporter of Brexit, and they would emerge from meetings with him grey-faced after a lengthy dissection of their alleged pro-EU bias. The former Today editor Rod Liddle likes to quote one BBC manager who said: “You do realise, Rod, that these people are quite mad?”

The more serious gap for the BBC is in its understanding of the world-view not just of Westminster Eurosceptics but of the voters driving them. The EU and its effects look different on a Doncaster housing estate or in a Lincolnshire village from how they are perceived in the W1 postcode area. Reporting from around the UK has improved but we know that London, where most network journalists are based, is one of the most pro-EU cities; and well-educated younger people, including most BBC staff, are much more In than Out.

I came across vanishingly few EU withdrawalists in my broadcasting life, which is at least one of the reasons why Michael Gove stood out when he was a reporter on Today. It is also hard to imagine someone with a vigorous commitment to withdrawal becoming chairman of the BBC, whereas Lord Patten’s past as a European commissioner was generally accepted. But Europhiles are not happy with the BBC’s coverage, either: they say that it is too often driven by the Brussels-bashing press and little time is spent explaining how the Union works.

This will be the trickiest of tightropes to walk in one of the defining votes for modern Britain. I do not believe that there is systemic bias. The BBC will be meticulous in allocating airtime for contributors and its journalists will display their characteristic professionalism – but they will also need to have some empathy with the opposing camps. There are decent and internationalist and patriotic people who conclude that we should leave the EU, just as there are those who want to remain in it.

There are other traps, too. Harding’s email notes, “Polls, as we know, can be unreliable”; and after the media were badly burned during the general election there will be an even closer scrutiny of how the headlines are shaped. We saw in Scotland in 2014 how the nationalists tried to build the story of the polls into their campaign: it was about momentum and a final surge towards the tape. In the EU referendum, the agenda might look very different if the campaigns were neck-and-neck with a week to go – as opposed to a 10-point lead for one or the other side. But can the polls now be trusted at all?

Related to that is the Westminster bubble’s obsession with process rather than policy. It is impossible to resist some forays in that direction when there is the David Cameron v Boris Johnson dynamic but the public is far less interested than journalists are in a Tory leadership campaign that could be more than three years away. Jobs, trade and security and a cool analysis of Britain’s place in the world will matter more in Truro and Caithness.

That brings a final challenge for the broadcasters: to bring this alive outside the news bulletins. When the future of the UK was at stake in the Scottish referendum, the response of BBC1 in its highest-profile network slots was feeble. There are signs – say, in the debate series just announced – that lessons have been learned. There is an obligation not to indulge in overkill and bore the voters rigid. Yet the importance of what is at stake cannot be overstated. We can manage without a few episodes of DIY SOS at 9pm on BBC1, and the opportunity is there to show that public-service broadcasters can facilitate a national debate that reaches everybody in a way that no one else can.

Roger Mosey is the Master of Selwyn College, Cambridge