“Do not adjust your set. Normal service from the BBC means you will hear people you disagree with saying things you don’t like. That’s our job.” A year ago that was my response to a warning (or was it a threat?) from the former culture secretary John Whittingdale. The BBC would face sanctions and fines from Ofcom unless we ended our alleged anti-Brexit bias, he said. My message was re-tweeted and liked by thousands.

So, it is with some sadness that I feel the need to reissue this reminder to those who I suspect were cheering me on then – ardent Remainers. Now it is Andrew Adonis who is trying to soften up the Corporation by claiming that the BBC is in breach of its charter, as we have – he claims, without presenting a shred of evidence – decided to back Brexit for fear of losing the licence fee.

He will, no doubt, have been encouraged by the column in the last issue of the New Statesman by James O’Brien, who chose not to condemn the BBC by name but to launch an assault on impartiality – the legal foundation of all broadcast news in this country – including at his own station, LBC. O’Brien claims that impartiality is all too often, in reality, bias, since it requires broadcasters to give “false equivalence” to those he claims are speaking objective truth and those making baseless assertions.

The former Tory chancellor Nigel Lawson, he implies, has no more right to be interviewed about climate change than someone who believes the Earth is flat. The last time Lawson appeared on the Today programme he got some of his facts wrong and we failed to correct him; it was a mistake we later corrected and apologised for. But if getting your facts wrong meant you could never again appear on radio or TV there’d be precious few politicians on air. Lawson has, by the way, appeared on Today to discuss climate change just twice in four years.

BBC programmes are not required to give equal airtime or weight to pros and antis in any debate. Our rules make clear that we have to deliver “due impartiality”. That word “due” makes clear that programme teams can and do make judgements on the validity of stories, challenge facts and figures and acknowledge that different people speak with different levels of authority.

Ah, I hear you say, but what about Brexit? O’Brien argues that presenters should act as tribunes of the people: any interviewee claiming that leaving the EU will allow Britain to “control our borders” should be “hit over the head repeatedly” with the “fact” that we already do control them, “until they surrender and withdraw”.

Let’s put aside the entertaining hyperbole and ask whether that would be the basis for a fair interview rather than the theatrical and entertaining confrontations that make his LBC talk show so enjoyable. You may regard the free movement of EU citizens as a good thing or a bad thing, but the claim that Britain controls its borders even when people can freely come here from 27 countries is just that – a claim or a debating point. It is certainly not a fact. Nor is it a fact, as O’Brien argues, that Britain’s sovereignty is unaffected by our membership of the EU. Indeed, the argument has focused on just that debate ever since Enoch Powell teamed up with Michael Foot to oppose our membership of the European club on the grounds that – you guessed it – it compromised our sovereignty.

O’Brien’s impassioned rhetoric cheers Remainers up. And Leavers can always listen to Nigel Farage on the same network. But we know where this sort of journalism would lead us if it spread beyond radio talk shows and phone-ins. You would end up with Fox News or a British equivalent.

Rupert Murdoch dreams of tearing up the law on broadcast impartiality here, just as it was torn up in America in the Reagan era. What was known as the Fairness Doctrine allowed opinionated presenters – as our law allows for James O’Brien – but ensured that a single network could not broadcast from a single perspective, day after day, without presenting opposing views. The doctrine was scrapped on the grounds that it “restricts the journalistic freedom of broadcasters” and “inhibits the presentation of controversial issues of public importance to the detriment of the public”.

So, the United States is now a democracy in which there are no shared facts and are only disputed opinions. Two-thirds of right-wingers watch Fox for their “facts” while liberals watch CNN, MSNBC or the old terrestrial networks to get theirs.

The problem this posed for public life in America was obvious long before the election of Donald Trump. When Barack Obama tried to open a debate on healthcare reform, Fox News said he would introduce “socialist death panels” in which government bureaucrats would decide who lived and died. On the other side liberals filled MSNBC with claims that Republicans wanted to kill the poor.

Impartiality is difficult. Perhaps never more so than in recent years when deep divides have opened up over Brexit, Scottish independence and inside both our major parties. We don’t always get it right. However, there is still a powerful case for impartial broadcast journalism that seeks to inform rather than influence, or sway, or respond to commercial imperatives, staffed by people who – regardless of their personal background or private views – are committed to delivering what Watergate reporter Carl Bernstein calls “the best obtainable version of the truth” and offering their audience a free, open and broad debate. The alternative is news that largely broadcasts people you like saying things you agree with.

I say as gently as I can to people on both sides of the Brexit argument – be careful what you wish for.

Nick Robinson is presenter of Today on Radio 4 and a former BBC political editor.