Frankly it could go both way! But let's say that I drew a lot of 'what if' kaigaku stuff that more or less fits into the story without disturbing canon, so these would end with his death ...
But this in particular is, definitely an AU, so we could be self indulgent and have Kaigaku redeem himself 😤 never turning into a demon at all, or managing to be cured, or living as a 'peaceful' demon like Yushirou for the rest of eternity 😤
I like the idea of him being a lifelong partner of Yushirou, i pondered this scenario a lot. Redeeming himself by living a long life and making sure the history of Kisetsutai is preserved - Kaigaku getting a sense of duty, a purpose that isn't entirely self centered if you will - in repentance for his selfishness and the trouble he caused (bc Grandpa still died in this I assume). So he was spared, and was gifted life when he should have died, but is cursed to see Zenitsu and everyone else grow old and die and carry the burden of their memory...😤
That or he stays human and Grandpa is fine and Kai grows old with Zenitsu and they kiss a lot 💕💕💕💕

....Or Zenitsu does kill Kaigaku as in canon, and lives his married life with Nezuko haunted by his failure to save Grandpa and Kaigaku. If I knew how to write it's a fanfic I'd love to write...
It can be a post canon scenario or written in the context of my Kai replaces Ino AU if you want more proximity between the zenkai, and a Zenitsu who must have felt very close to put Kaigaku on the right path but ultimately failed anyways...
So in both cases, Zenitsu grows older, and marries Nezuko. He wears the black uroko haori and keeps a magatama on a blue string as a memento. He loves his wife, but when he looks at her, he sees a woman who turned into a demon and was eventually cured. And he can help but sees through her, the face of a boy turned into a demon that He failed to save, and never had a chance to try and cure. Time passes, and the sight of his wife becomes unpleasant, because it mirrors his failures. His relationship with Tanjirou deteriorates, as Zenitsu's hard earned self confidence is slowly replaced with self consciousness. Perfect man Tanjirou, who saved his sister through pure sheer will, never wavered, never complained, is left with one eye and one arm. And he still doesnt complain, savouring every day of his hard earned peaceful life, working hard at housechores and cutting wood with his only functioning hand...
If Zenitsu used to admire him, little by little he starts to become really irritated. Because Zenitsu can't bring himself to work hard, can't bring himself to wake up in the morning, and cant even bring himself to look at Nezuko in the eye. He just lives his life progressively alienated from everyone else, haunted by the ghost of Kaigaku and all his failures....😚😚
.....that's a depressing story!!!! Maybe kaigaku can live and smooch zenitsu actually.