But that's precisely it, they're outcasts of kitsune society and can't be accepted by the Frostwood but they've picked up bits of the Frostwood by virtue of living there. That puts Kinu in the perfect position to be at the top, at least of that little microcosm. Keep in mind, we don't want to draw the attention of mainland Kitsune society anyway which is why Kiyoko doesn't take a more involved position so Kinu likely won't go any higher than being an aide to Komari and then to whichever sister takes over if Komari gets to pass on (not accounting for divine shenanigans or other plot developments). If she wants to nudge the collective consciousness of the kitsune society in the tree, she'll be better able to do so from a position of respect and authority than without it.
I agree the kitsune society is kinda ass but the tree being an outcast enclave where the rules are more relaxed helps things out. But perhaps most importantly, all those rules and stuff are apparently because kitsune need to restrain themselves, because they're "soul suckers" and without that restraint they could so easily become monsters. That's why you're still an outsider even after Keros names you their champion, you don't have the hunger and you didn't have to grow up with it, it's not a fundamental part of your existence. But for Kinu it is, and that society is what she spent her entire life learning of and living in a little bubble of. Whether Inari or Hime, she can't get away from that. Indeed, Inari really gripes about the "princess" bit of "kitsune princess" because it's not like she can get away from the "kitsune" part.