Re the OP, Pétain makes an interesting comment in the section comparing French and Prussian practice. This translates VERY approximately as:
"The splitting of the battalion into columns of companies, each of them enjoying a certain independence, is a feature of the Prussian regulations. The French battalion columns of division attempt to imitate it, but only succeed in theory as their use remains absolutely rigid.
The independence permitted to the Prussian company, which becomes the unit of maneuver and combat in some cases, is a characteristic superiority of the Prussian regulations, although this measure alone is not enough to explain the constant success of Prussians.
What, then, accounts for the superior maneuverability of the Prussian infantry?
The Prussian army has a brain represented by his Grand Staff and its War Academy. In addition, its training is directed with a view to war. Indeed, while the French infantry maneuver on the parade ground, the Prussian infantry is used to frequent training in the fields around garrisons. In addition, Prussia had long ago instituted major annual maneuvers accompanied by marches and camps, which provide the Prussian infantry with an incredible flexibility in maneuver over other armies. The practical training in different terrains, in conditions close to those of actual war, inculcated in units the ability to think and the qualities of initiative which were clear from the first contact French army. There is no need to look for another cause for the great maneuverability of Prussian infantry."
So Pétain clearly recognises that the Prussians had a greater flexibility below battalion level.