Honestly, it makes sense that Michikatsu is the way he is.
Lived at a time where the value of a man was defined by strength, ability, and usefulness
Raised by a man who prioritized strength, ability, and usefulness
Mother was barely in the picture for him
Physically abused by the only parent who had a real hand in raising him
Developed his identity around protecting his baby brother
And then, all of sudden, his brother shows such natural talent that he leaves veteran samurai in the dust. Suddenly, Michikatsu is nothing in the eyes of his father — the only parent he's ever known abandoned him, just like his mother. Suddenly, his entire identity of being Yoriichi's protector is shattered, because Yoriichi never really needed him, did he.
And Yoriichi? He doesn't even want this power. Yoriichi leaves.
There aren't very many feelings worse than knowing you're only chosen by default, because there are no other options.
And he hates himself, and he hates Yoriichi — Yoriichi, who never complained despite his father's neglect, who never envied his elder brother. Perfect Yoriichi, untouchable. A god among men.
And then he gets the slayer mark, and finally, maybe finally, he can be as good as Yoriichi.
And then the marked slayers start dying, never surviving past twenty five. And Michikatsu is only a few months away from turning twenty five, and he still hasn't even begun to master Sun Breathing, only the inferior Moon Breathing, and he's nowhere near where Yoriichi is, and he's going to die a failure —
Until Muzan gives him an offer. Immortality. Power. The ability to finally become Yoriichi, become better than Yoriichi. Better than perfect. To not die a failure.
Really, how could he refuse?