Stop quarrelling, it was a mix of what you both said.
Since May 10 events just rushed too fast for even the British (least the sclerotic French commanders) to grasp long term implications. In August 1940 Dakar was another tragedy born out of despair and miscommunication.
Just try to put yourself in the head of a military Frenchman in the summer of 1940
Warning: strong headaches ahead.
Bluntly: French commanders were completely and entirely at lost. Some rallied De Gaulle, other rallied Vichy on behalf of Spain-like neutrality (let's get out of that confusing war) and others were actually pro-Germans, but not Nazis, and some others were far right, and pro-Nazi.
There was also WWI and pacifism - Germans had equally suffered, after all. Same death toll. But the German army of 1940 was not the one of 1918 or 1914. There was that Nazi ideology that did not existed in WWI.
Don't you think things are a little messed up ? clearly they were.
In the end France had hit a situation where one has to decide by himself which side he wanted to fight with. But individual thinking in late 30's France wasn't a walk in the park. There was a crapton of ideology and politics flying everywhere.
A striking aspect of those who early rallied to the Free French was the strength of their character, and this goes far beyond De Gaulle or Leclerc. Free France veteran testimonies are all strikingly similars. Chaos was such, rallying De Gaulle in London one needed quite a strong epiphany.
To survive as a French military in the confused year 1940, one has to think pretty fast and forward. Which is not given to anybody. Hence why Free France was so little early on...
Let's take a striking example. I know the story of a French fighter pilot shot down in May, evacuated to Britain at Dunkik, then two month later founding himself on the British side - in Dakar - and then, when he landed at Dakar airfield as a scout to try and negociate, he found his former squadron there - now the ennemy since passed on Vichy side ! And he was taken prisonier. By fellow pilots. Go figure.
The British were in shock at France rapid collapse, Vichy France didn't existed (yet) and it could go either Franco or Mussolini way - with or without the Axis.
Although Sea Lion seems absurd in retrospect, as of July 1940 it was not, and this was the root cause why Britain feared Germany taking over the French fleet. Kriegsmarine + Regia Marina + French Fleet = big challenge to the RN.
One should see MEK as France own Tarento - as said, elimination of a powerful fleet in being threatening the Mediterranean.
The Navy was clearly the best prepared and equiped of all three French armies, far better than either the Army or Air Force.
If they do not press to Toulon, the only other place is first Dakar (and you what happened in August 1940 !) then the Caribbean. Or the ships could be interned in Alexandria - Lorraine actually was interned there, then handled to the Free French in 1943.
Which lead us to Some Bloke splendid TL "The Mers El Kebir resolution".