I've a possible puzzle piece that could explain Pochita/Chainsaw part of the mc quite well, thinking about all of the quirks and properties it has.
Demons are born of feared things and live in an endless cycle. Chainsaw(the man being a recent overworld addon) is also a demon and assumedly born as such. CS kills demons and is majorly feared by them. Demons die and loop in a reincarnation. CH excises this. CH doesn't die properly, so it might be the thing it has jurisdiction over, like all the other demons have over some feared(and revered) concept. As stated earlier, chainsaw as a thing was created as a tool for childbirth, which I suspect matters quite a lot conceptually.
Chainsaw might just be the fear of ceasing reincarnation. endless death. Whatever analogy you prefer.
Demons have few things to fear (even death has been used as a relatively palatable option to avoid other things). They, however, seem to fear the idea of hell and back not happening and the doors (one of the only times we have seen truly frightened demons). This chapter also explicitly stated it is the most feared thing. My suspicion: Chainsaw is a manifestation of the greatest fear demons share.