Yes, he does need it and I believe he has it. However that doesn't mean he will utilize it to it's fullest.
I think the reason people might question it is Mihawk doesn't seem to embrace his CoC. His personality and his innate CoC-ability don't gel, and that is perhaps the reason he is one of the strongest in the world instead of the undisputed strongest in the world now that he is in his prime and the main rivals are gone (Shanks lacks an arm, WB dead, BM/Kaido presumed dead). I see mainly 3 different types of conquerors in the series (doesn't fit perfectly but you get the main jist of it). Belonging to one group of conquerors don't mean all are as successful in their respective field (looking at you Kidd), it's just their different styles. Some could be swapped (Sengoku for example).
Conqueror of Conquerors
Luffy, BB, Roger, Joy Boy, Kidd, Imu, Rocks - They want to conquer the world including other conquerors in order to shape the world according to their beliefs. They lead other conquerors.
Conquerors of Areas/Domains/Fields
Shanks, WB, Sengoku, Hancock, Mihawk, Zoro, Oden, Ace, Katakuri, Rayleigh, Gorosei, Garp - They seem content to conquer a part of the world, or a certain field, and then live in peace. They rarely lead other conquerors.
Conquerors that Scheme
Kaido, BM, Doflamingo, Chinjao - They conquered a base but couldn't conquer other conquerors, instead using list and scheming in order to take down their rivals. They don't want other conquerors below them believing one leader is enough.
For all we know at the moment Mihawk seems to be a lesser version of Shanks & WB (CoC-wise). He has conquered his field of the world and now wants to live in peace. Due to his lone wolf personality he can't even (or doensn't care for) attract a family/nakama like Shanks or WB. That's different than the highest level of Luffy, BB, Joy Boy, Roger, Kidd, Imu, Rocks.
Using other wording Mihawk said so himself during Baratie, the PK conquers it all not only a field. And Chinjao said the same, pointing out BB has what it takes to conquer even other conquerors. Contrast that with BM feeling threatened by her one conqueror Katakuri, Kaido saying he doesn't want conquerors or WB saying he doesn't feel Oden will be satisfied playing second fiddle. Not until he was ready to hand over the reign did he actually take on another conqueror in Ace.