My guess would be that Crocodile, having himself some military strength, network and the necessary intelligence, contacted Buggy first to discuss the establishment of such an organisation (knowing Buggy has the network and ressources too) and after that got in touch with Dracule, revealing his plan and asking whether he, Dracule, who is excited about being in the piracy game again, would be excited about participating in chasing down the marines systematically and being rewarded for it, rather than being the one targeted, effectively flipping the roles.
The fact Mihawk lets Buggy call himself captain is proof Mihawk either isn't a conqueror, or if he is, he isn't a very good one.
With all due respect, but true conquerors have proven to swollow their pride.
- Shanks being beaten up and spitted on by fodder Bandits
- Luffy bowing down to Hancock to ask for her subordinates to be spared
- Zoro bowing down to Mihawk to receive teachings, in order to be a good right hand man for Luffy
- Roger bowing down to Whitebeard and asking him to have Oden join him
- Oden embarrassing himself for 5 years and dancing, in order to try and make peace with Orochi
So why would you assume Mihawk would even care about the public perceiving Buggy as the leader, when everyone with a brain knows anyways that this is bullshit? Again, Crocodile himself in Marineford told Doffy that he won't even team up with him as equal partners, but rather only accept him as a subordinate. But apparently Crocodile in this case doesn't mind either to be perceived as an underling of this fodder clown. So obviously, both Crocodile and Mihawk will have reason to give a shit about public perception.