I think that the reason that the French have been so quick out of the blocks, is that North Africa is 'their' empire and I imagine that they will be damned to play second fiddle to Perfidious Albion.
The have been "interested" (not officially involved) in Chad for some time if memory serves well.
I also would not dcry either their military or their government's record in providing interventions as it suits their national interest.
There might not be any alliance boots authorised, but I cannot believe that there are not Legionaires on a border somewhere or more than able to deploy from Corsica.