The burden of simply being way too stronk...
I thought mihawk taught him how to deal with that
Mihawk taught him CoO and especially CoA. Zoros CoA is among the best we have seen in the series, Mihawk did really well in teaching Zoro that. Mihawk never taught him CoC or even told him that he possesses CoC.
Why? I suppose one explanation could be there was not enough time. The second explanation could be similar to what Sanji has to do next.
It's all good that they accept Luffy as their boss but as Zoro by accident figured out vs King he had to realize that in order to help Luffy he himself needed to become a King (of hell). Only a king can truly protect the king of kings. That is probably why King never became a king, he was content being #1 servant behind Kaido but lacked any ambition to become a king himself. If he makes a comeback I can see him becoming a king if his ambition is too become king of the lunarians.
Katakuri was also content being #1 servant behind Big Mom but he desired to become a king himself in order to protect his family, amongst them many weaklings. King never cared about any of the crew except Kaido who he viewed as unbeatable. Katakuri knew Big Mom was out of her prime and old. Zoro realized Luffy can't handle everything himself. Sanji will soon realize the same thing.