In the wake of the Warlord system dissolving, three former Warlords came together to form an alliance/organization, and while it’s meant to be an equal alliance the Marines have mistakenly labeled Buggy the leader.
Honestly I don’t think it’s particularly surprising that Crocodile or Mihawk have tried to correct this.
Crocodile seems prefer working in the shadows. He’s a manipulator and a mastermind. He probably sees value in allowing people to think Buggy is in charge to draw attention away from himself.
Crocodile is someone thats already faced defeat and been captured. However, he knows that Buggy, while weak, has an outstandingly amazing resume from being on Roger’s crew to masterminding Impel Down, not to mention so quickly entrenching himself into the underworld that as soon as Doffy’s years long presence vanished, Buggy was immediately poised to be the #1 player in the underworld markets.
It makes absolute sense for Crocodile to use Buggy as a figurehead leader to further promote and boost the organization up. Crocodile as a captain doesn’t quite have the same sort of impact. Buggy does, at least to most of the world. People like Kid can see somethings wrong, but the general public won’t.
Mihawk is a little more inscrutable. His motives have always been somewhat mysterious and he seems to be at least partially driven by whatever passing whim catches his fancy.
Why he joined Cross Guild in the first place is hard to say. What we can say though is that Mihawk doesn’t seem the leader type. He’s one of the few Warlords we were introduced to without a crew or organization attached to him. So his reluctance to lead could be why he doesn’t feel a need to clarify Buggy’s position.
I guess that Mihawk decided to join just for the fun, Mihawk feels like he just quickly got bored cause the marines didn’t send an admiral after him, so this seemed funner.