What nobody appeared to be aware of - least of all the interviewer Stephen Sackur - was that Royal is one of the most undemocratic politicians in Europe. Not only was she roundly beaten by Nicolas Sarkozy in the 2007 presidential elections, but she has lost every significant poll since.
The Right-wing Sarkozy - a deeply divisive figure now involved in numerous corruption enquiries - won by more than six points in 2007, making a mockery of Royal's claim to be the natural choice for head of state after more than a decade of conservative rule.
She was defeated when she tried to become First Secretary of the Socialist Party in 2008, and yet again in 2011, when she achieved just seven per cent of votes in that year's presidential primary. Royal's greatest humiliation came in 2012, however, when her opponent in the parliamentary seat of La Rochelle beat her with 63 per cent of the vote.
Many believed that 2012 would be the end of Royal's disastrous late career, but her ex-boyfriend, Francois Hollande, saved her dignity. With characteristic ineptitude, the President appointed the mother of his four children to his government without any consultation with the French people whatsoever. The pair are meant to be democratic Socialists, but - displaying breathtaking arrogance - they saw no reason why electoral failure should be an impediment to power.
This did not stop Royal telling Sackur that it was Britain that was entering a "fantasy world", and that "what we need is clarity and truth, and to apply democratic principles".
Whatever your views about Britain's status in Europe, Royal's BBC performance encapsulated the cloud cuckoo land madness surrounding the EU debate. Not only have facts been routinely manipulated, and indeed ignored, but ludicrously misplaced foreign commentators are promoted by Britain's state broadcaster with next to no examination.
Democracy is anathema to Royal. What she represents is an outdated and anxious Paris establishment that is prepared to scorn democratic politics in order to maintain its gilded position. She is supported in this by plenty of colleagues, including the unelected Economy minister Emmanuel Macron who has warned that Brexit will reduce Britain to "a little country on the world scale".
It was Macron's turn to appear on the BBC as recently as April, when he told Andrew Marr that Britain would be "completely killed" in global trade negotiations if it left the EU, because it would have no more significance than Jersey or Guernsey. Macron, a former merchant banker, is now feted as a future president in France, despite never having contested an election in his life. Once again, Marr chose not to mention the latter fact.
Like all French politicians, unelected Secretaries of State in Paris are not subjected to anything like the scrutiny of their Westminster counterparts. Beyond pay and expenses packages worth the equivalent of upwards of £15,000 a month, Royal and Macron get other perks such as a chauffeur-driven car, and free state-owned apartments.
When appearing on their own country's television programmes, they are treated with the reverence of minor royalty. Royal is a regular on all channels, pouring out the kind of unedited and unchallenged platitudes that she offered Mr Sackur for what often seems like hours on end.
Many of the 17 million British people who voted to leave the EU did so precisely because they believe fat cat politicians just like Royal run it. A project that was started by France and Germany has now become so bloated and unaccountable that it is inspiring a class of politicians who have completely forgotten what the word "democracy" actually means.
Royal, who at 62 should be old enough and wise enough to know better, even pledged to prevent the "contagion" of democracy, telling Sackur: "But now the vote has happened, we must send a signal to other countries who might be tempted to use the vote politically." More sinister still, she even discussed the possibility of "punishing" the UK for its Brexit vote, before deciding - apparently with absolutely no sense of irony whatsoever - that this "wouldn't be consistent with my idea of democracy".
