I finally found an authority for the notion of who was supposed to be mounted in a British infantry battalion. From "General Orders and Observations on the Movements and Field Exercise of the Infantry," 1804:

"The Field Officers and Adjutants of the battalion must at all times be mounted, in order the more readily to give ground in movements; speedily to correct mistakes; to circulate orders; to dress and preserve pivots when they ought to cover in column in a straight line, or when entered on an alignment, on which they are to form; and especially to take care when the column halts, that they are speedily adjusted before wheeling into line. These operations in the movements of the battalion, when connected with others in line or column, no dismounted Officer can effectually perform; nor in that situation can he see faults, or give aides which his duty requires. The direction of lines of formation and movement must be taken and given by the mounted Field Officers and Adjutants of the infantry. Where a brigade or a considerable body is acting, camp colours are improper and insufficient for this purpose; and the horse's head is the object which the pivot flank leaders in column pass close to on their outward hand. FOr the accurate performance of this important point of duty, it is necessary that the horse should be steady, active, and well trained, as their riders must be alert, intelligent, and well grounded in the true principles on which the different manoeuvres are executed."

Cheers - Howie