The BBC is often bad at defending itself. As the nation’s crucible, upholding an idea of fair reporting in the turmoil of these bitterly divided times, its journalism comes under ferocious and unjustified attack, so it was good to hear its chairman David Clementi standing up for its journalists at the Royal Television Society convention on Wednesday.

He protests at politicians who give their approval for increasingly “explicit and aggressive” abuse directed at BBC journalists. Politicians “stand by and watch”, giving tacit support to hecklers at press conferences trying to intimidate BBC reporters asking tough questions. From right and left, pro- and anti-Brexiteers in the red mist of increasingly ferocious national debate, hard questioning by BBC journalists is taken as proof of bias.

The first female political editor, arriving at such a fraught time, was bound to attract most abuse, just as female politicians and journalists of all stripes draw a particularly vicious strand of misogynist hate. How shocking that Laura Kuenssberg has needed a bodyguard. Despite the murder of Jo Cox and death threats against MPs, particularly against women, this sharp and incisive questioner hasn’t buckled.

Walking the BBC line of unbiased reporting is becoming harder by the week, as both leave and remain supporters tear apart every word of every story. Nor does being attacked by both sides guarantee that they always get it right. But on Brexit questions of fact versus fantasy, how are BBC journalists supposed to balance reporting on actual bad effects happening now – the tumbling pound, finance HQs moving to EU capitals, Euratom and the medicines agency departing, vanishing EU nurses, imminent chaos at the ports – against vanishingly few putative Brexit benefits? BBC reporters are not obliged to split the difference between flat-Earthers and round-Earthers – or, these days mercifully, between climate deniers and climate change evidence. Making judgments on the likely truth of what they report is part of their job.

Mounting abuse of the BBC could in the end destroy it: it only survives on the trust and affection of most citizens. Those on the left joining in the attack, dismissing the BBC as part of an “MSM” plot, fuel the right’s aim to dismantle and privatise it. Murdoch and his press seize every chance to attack it, mainly for commercial reasons. He has always argued for the BBC to lose the licence fee, and to become a small subscription service. Today his bid is referred to the Competition and Markets Authority: if he takes over Sky, he will push next to abolish laws obliging broadcasters to be unbiased, Foxifying our news networks. Where does that leave the BBC?

The political animus against the BBC runs through the right of the Tory party, indignant at the very existence of a publicly funded organisation that is phenomenally successful, its every triumph an affront to their anti-public service dogma. At every opportunity, they chip away at it: George Osborne sliced away a third of its funding. Its governance is frequently reinvented, its charter renewal and licence fee rattled as a threat. Successive governments lean on it politically, mainly but not exclusively Tory: Blair blasted it over the Iraq war, while Harold Wilson boycotted it. But usually the bullying comes from the right: recently Liam Fox demanded a meeting to protest that the BBC was putting out anti-Brexit propaganda. As a national beacon, the BBC is first to be attacked for national failings others share – paying its women less, over-paying its managers, or on Wednesday from Ofcom for its “woeful” lack of diversity on and off screen. It has a tough moral duty to be best at everything, so it’s bound to fail sometimes.

All polling shows the ardent support the BBC still commands, one of the few unifying icons of national pride, along with the NHS. The envy of the world, its news is trusted as no other. But here’s a perplexing question: if we have this great national broadcaster, why is British political knowledge and understanding so weak? Shouldn’t we have the best-informed democracy? Why has the BBC not had a greater measurable cultural and educational impact? Polls show consistently how little people know about the basic facts on which to make political judgments.

But that may demand too much, as if the BBC could correct all the fundamental ills in the society it reports and reflects. Even now a supposedly declining press has a stronger pull on the nation’s psyche than our national broadcaster. The BBC is still a weak voice to rebalance a political landscape grossly distorted by a dominant far-right press for the near-century of the BBC’s existence. That’s all the more reason to defend its journalism against raucous assault in a country more viscerally divided than ever before.

• Polly Toynbee is a Guardian columnist