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I don't know if anyone else has already posed this theory but I assure you I came up with it on my own.

I believe the Chinjao Family are a part of Kaido's crew and their True Objective is to punish Doflamingo for losing Caesar and the SAD production facilities.

I came to this conclusion because of the numbers tattooed on the Chinjao Family's bodies. They'd bothered me since the first time I saw them but I never had a theory until now. I was thinking that they could be positions/ranks in their family, but that made no sense since Don Chinjao is the boss of the family but his number is 12(?), it should be 1 or 0 for the boss, or even no number at all to signify that he isn't included in the ranking. Nonetheless, I still felt that the numbers are a ranking system/position identifier for some crew/organisation. Then it hit me:

Kaido's crew. And immediately I knew it had to be so. That would mean that their true objective is Doflamingo. It would make sense to send them, seeing as Don Chinjao is a veteran with legendary strength. Only afterwards did I remember that Scotch, Kaido's cyborg subordinate, also has a number tattooed on his body (18?), supporting my hypothesis.