"I watched this video and despite a huge cast of french historians it fails to examine the key events of 16thJune when Reynaud resigned and Petain was requested to take-over. I find this surprising because I have a book in front of me written by Benoist-Mechin, a member of the Vichy Government who seems to describe the events accurately - or a least one that tallies with the Minutes of the British War cabinet."
I watched the video again completely and as it is some years ago that I watched it, I see now again how good it was.
I see now that after the film there is a postcriptum that is best not watched as it seems to me a bit jingoist not to compare with the documentary.
If you look to the video from 4:56 the whole story is there about the misunderstanding of Churchill by Paul Baudouin.
I found it back in the wiki, but I read it elsewhere too, if I recall it well, in William Shirer's Collapse of the Third Republic...
From the wiki:
"What would prove to be the final meeting of the Anglo French Supreme War Council took place at the Préfecture in Tours on 13 June. When Spears arrived, the British delegation – Churchill, Lord Halifax, Lord Beaverbrook, Sir Alexander Cadogan and General 'Pug' Ismay – were already there. The French Prime Minister, Paul Reynaud, was accompanied by Paul Baudoin, a member of the War Committee. Spears found the atmosphere quite different from that at Briare, where Churchill had expressed good will, sympathy and sorrow; now it was like a business meeting, with the British keenly appraising the situation from its own point of view. Reynaud declared that unless immediate help was assured by the USA, the French government would have to give up the struggle. He acknowledged that the two countries had agreed never to conclude a separate peace  – but France was physically incapable of carrying on. The news was received by the British with shock and horror; Spears' feelings were expressed by the exclamation marks which he scrawled in his notes. Spears noted Churchill's determination as he said, "We must fight, we will fight, and that is why we must ask our friends to fight on." Prime Minister Reynaud acknowledged that Britain would continue the war, affirming that France would also continue the struggle from North Africa, if necessary – but only if there were a chance of success. That success could come only if America were prepared to join the fray. The French leader called for British understanding, asking again for France to be released from her obligation not to conclude a separate peace now that she could do no more. Spears passed a note to Churchill proposing an adjournment – a suggestion that was taken up.
After the meeting, de Gaulle told Spears that Paul Baudoin had been telling journalists that Churchill had said that "he would understand if France concluded a separate armistice" ... "que l'Angleterre comprendrait si la France faisait un armistice et une paix séparée". Spears realised there had been a linguistic misunderstanding. When Reynaud spoke (in French) about a separate armistice, Churchill had said, "Je comprends" (I understand) in the sense of 'I understand what you say', not in the sense of 'I agree'. Just as Churchill was about to take off for Britain, Spears obtained his assurance that he had never given consent to a separate armistice. But the damage had been done and, on 23 June, the words would be quoted by Admiral Darlan, who signalled all French warships saying that the British Prime Minister had declared that 'he understood' the necessity for France to bring the struggle to an end'.
Shortly before lunch a telegram arrived from London agreeing that France could seek armistice terms provided that the French fleet was sailed forthwith for British harbours pending negotiations. Spears and the Ambassador felt this would be taken as an insult by the French Navy and an indication of distrust. Reynaud received the news with derision – if Britain wanted France to continue the war from North Africa, how could they ask her fleet to go to British harbours? He had spoken by telephone with Churchill and asked Spears to arrange a meeting with the British Prime Minister, at sea somewhere off Brittany. The meeting, however, never took place as he preferred to go in a French warship and this never materialised. As the day wore on, Spears became more aware of defeatism – but the hard-liners tended to be socialists. His British uniform struck a false note and people avoided him.
On the afternoon of 16 June, Spears and the Ambassador met Reynaud to convey a message from London – it would be in the interest of both countries for the French fleet to be moved to British ports; it was assumed that every effort would be made to transfer the air force to North Africa or to Britain; Polish, Belgian and Czech troops in France should be sent to North Africa. While they were arguing with increasing acrimony about the fleet, a call came through from de Gaulle, who was in London. The British proposition was nothing less than a Declaration of Union – 'France and Great Britain shall no longer be two nations, but one Franco-British Union. Every citizen of France will enjoy immediate citizenship of Great Britain; every British subject will become a citizen of France.' Spears became 'transfixed with amazement'; Reynaud was exulted. When the news got out, hard-liners such as Georges Mandel were pleased and relieved. The proposal would be put before the French cabinet. Spears was optimistic that it would be accepted for how could it be that of the countries fighting Germany, France should be the only one to give up the struggle, when she possessed an Empire second only to our own and a fleet whole and entire, the strongest after ours in Europe'. Yet he joked that the only common denominator of an Anglo-French Parliament would be 'an abysmal ignorance of each other's language'! 
While the cabinet meeting was taking place, Spears and the Ambassador heard that Churchill, Clement Attlee, Sir Archibald Sinclair, the three Chiefs of Staff and others would arrive off Brittany in a warship the next day at noon for talks with the French. However, the French cabinet rejected the offer of union; Reynaud would be resigning. One minister had commented that the proposal would make France into a British Dominion. Spears, on the other hand, felt the rejection 'was like stabbing a friend bent over you in grief and affection'. Churchill and his delegation were already in the train at Waterloo station, when news of the rejection came through. He returned to Downing Street 'with a heavy heart'.
peccavi, on the French fora I heard a lot of critique about Jacques Benoist-Méchin...
Kind regards, Paul.