The Centurion and his horse questions.

[jkaler48]

Of course it an assumption that Centurions are involved in counsel but they are the Commanders most experienced officers.

In Gallic Wars, Caesar mentions more than once about strategy meetings involving officers and centurions.

That's good input. Centurions or at least Senior Centurions, perhaps the cohort commanders, may well have consulted with the Legion commanders when necessary while on the march.
Of course while it may have been more difficult to do so if they were on foot rather than mounted it still doesn't amount to proof about the base question about Centurion's having been mounted on the march. The Legions may not have considered it as much of a difficulty as we might.