Adding to this in the eyes of the United States and a good deal of the French people it will give the Free French a greater degree of legitimacy and also recruits as well may even cause the more rapid defection than OTL of the French Middle Eastern and African to DeGaul.And don't forget the USA was sending food convoys to Vichy France. Because they were now out of the war and were a "Neutral"! If they now allowed the Germans to use their own fortifications against the British Commonwealth Forces, then they become a combatant. And that would not play out well for France in the USA! Plucky and defiant Britain forces the dastardly Nazis back and the French allow them to use French fortifications? Whose side are those French on, anyway?