The necromancy lore page at UESP seems to indicate that a soul IS required to animate a corpse.
Typically, this soul manipulation is accomplished by binding a soul to a physical form which has been prepared by the necromancer.
Necromancers actually prefer that the original soul has fully departed the body. See this extract from On Necromancy:
As to freshness, be careful in this consideration as well. Have you ever wondered why there are so many skeletons among the reanimated undead, fewer zombies, and only a scant few revenants? The longer a body remains inanimate, the less hold its original owner has on the corpse. A spirit can stay tied to its remains for days, weeks, or even years—the shorter the time, the more likely the spiritual umbilicus exists.
A wise necromancer does not wish to fight for control of his creation with an angry spirit seeking a way back into the world. Best to be certain all of a creature's soul has departed before reanimation begins. Even should the necromancer win the battle, it is a cruel victory, tormenting a spirit on its way to rest.
Raising the dead so recent that the soul has not yet fled is ill-advised, as true resurrection is not the purview of the necromancer, but something best left to gods and priests.
Necromancy appears to be related to the other forms of conjuration, summoning Daedra and Daedric weapons. A ritual is typically performed to perform necromancy. From Worm Saga:
Thus found I new allies to study the death-rites, the sacrifice rituals, the summons of ghost-wights.
Some lore even explicitly mentions the benefits using bound Daedra as the source of the reanimation. From On Summoning Skeletons:
Plus, if you use minor Daedric spirits to animate them, corporeal undead have no sense of self or memory of past lives.
So I guess in conclusion... Since it requires a soul to reanimate a corpse, it makes sense for the reanimation to performed with the help of a soul gem. Since it is preferable to reanimate a corpse with a different soul than the original, it makes sense for it to be performed on a corpse regardless of whether its soul was trapped. However, these conditions are already satisfied in-universe with the idea that the reanimating soul is a bound, summoned soul (possibly Daedric).
If you prefer to think of the reanimation provided in this mod is a new type of ritual that doesn't require dabbling in the dangerous and corrupting area of black soul magic, since you only use the white souls of lesser beings, then it could be a fun roleplay thing!
