Their martial training is primarily for their own benefit, they use it to improve themselves and cleanse themselves with the end goal of becoming an ascended.
Their job is ferrying souls past the Veil, not being a military.
Not just an official army, its their own and only purpose for existing at all in this cosmic machinery.
If we call the Kyrians a second standing army just because they a fighting force, then might aswell proclaim all covenants an army as the Venthyr got the Stone Legion and the fae have the Wild Hunt.