Currently (Apotheosis 8.3.4 in Minecraft 1.21.1, Craftoria 1.20.3), when a non-pillager (and non-player) mob uses a crossbow, it is not shown using a crossbow the same way a pillager uses one; instead, the crossbow goes through its loading and shooting animations while remaining fully in the mob's main hand, without the position of the mob's hands changing.

Would it be possible to make other humanoid mobs go through similar poses as players and pillagers when using a crossbow?

They are animated under normal conditions. Iirc craftoria has a mod which is replacing and reanimating many of the entities, which is likely incompatible with my support.