Problem is that Drake also said in the same chapter he saw CP0 in the Wano capital and Rear Admiral Koby was shocked to know. CP0 are even strengthening Kaido's ally by giving him warships. back then we saw Akainu criticizing the blonde Gorosei for leaving matters to CP0 to solve which further indicate that they don't corporate together.
1. In Dressrosa CP0 never interacted with Fujitora, they backed Doflamingo.
2. In WCI they were part of the underworld ( and marines only knew about what happened from Morgans' newspaper )
3. in Wano they were dealing with Orochi
they are absolutely corrupted organization under the CDs only and they never helped the Marine to neutralize evil threats thus far.