I'm starting to feel argumentative, but I suppose this "pacing yourself" thing doesn't apply to men in mail hauberks, carrying kite shields and six feet of spear, or men in full plate marching along with two-handed swords over their shoulders?

How about soldiers carrying those very billhooks you mentioned? They walk at a perfectly average speed.

Any way you look at it, singling out halberdiers and voulgiers as exceptionally slow-moving units is devoid of logic.