The fact that they do actually have machine spirits of a sort, is what I think makes the most sense to relate to Shinto. They were also constructed for a specific purpose, and serve an immortal empress. These are unique things about their culture that can be used to culturally unite them...
*That said*, I am very wary myself about Yamatai soldiers turning into screaming zealots, or weirdos who use unseen ghostly Kami to act irrationally. There are war gods in Buddhism, too, believe it or not.
This is why I propose that the chief function of ritual in this kind of shamanism still existing in various levels, is purely as a self-actualisation thing.
A Miko *idolises* a Kami, serves a vessel's *spirit*, but the Megami and the Captain are still the *literal* embodiment of that excellence. It's about self-improvement, not arrogance or navel gazing.
Having something to cling to when half the ship has been eaten by alien monster-squids... It's worth having. Pride only works if you always win.