I sort of feel the weapons of the Crucifixion should be weapons for the fight for good against evil, rather than the fight of order against chaos. Sure, Heaven doesn't want the Gates to fall, but the fight there has been about Aesir, Fae and Wizards... while Heaven fights on the field of angels, demons, and souls.
As such just plain healing, or otherwise making a human able to reach the full potential of humanity, seems like a reasonable power for such items. If it must be a magical cure, curing black magic taint seems more reasonable, but then it would feel a bit duplicative of the blackstaff.
Perhaps it really is more of a ward against supernatural threats; when those threats are disease-based (as Nic did with the fake one) it will cure, when they're more direct, maybe it works more like Harry's shield bracelet.