From what I can recall, Achilles' heel of French Air Force was very similar to their Armoured Forces one: organization.
As far as I know, the French Air Force was subordinated to the ground army, which commanded them in small groups attached to the ground divisions individually, which were managed by a strict bureacracy far away from the frontline.
This produced a very innefficient behaviour, because each tiny piece of the Air Force was "reserved" by the ground officer to meet their division's needs, and forced them to fight in disadvantage most of the times.