Don't forget, Yamato was cuffed with seastones, the flyers would have it easier to caught her since Yamato was nerfed asf with the seastones. We know even haki get effected by seastones, no devilfruit power and also limited movement. And yet Yamato was still a big threat to caught up.
im doubt that the flyers would get send out if King would know that Luffy break Yamato seastones.
Flying Six mission: Caught a nerfed Yamato with limited movement, less haki power, no df and with seastones on her hand.