And if Big Mom and Kaido take each other out, that's two less people between him and OP. By your logic, my way still ends up better.
Big Mom wants to keep the only guy who can make SMILEs possible for herself and her own research, or she'll murder him. That alone is enough reason for Kaido vs Big Mom even without all the connections between them.
This doesn't necessarily suggest that Kaido doesn't know who Caesar is. It simply could be that he doesn't know Caesar was even kidnapped by them. Do we know that Kaido even knows that Caesar is in trouble?
If Kaido doesn't know who Caesar is or that he's in trouble, that's something Oda can easily fix by having news involving Caesar, Big Mom and the SHs get out, and having a subordinate tell him upon them seeing that news.
Big Mom has shown herself to be a woman who would kill her own daughter for ****ing up her plans and goals. She already had Bobbin destroy an entire town because they wouldn't give her treats, and threatened to destroy all of Fishman Island over candy. You really think Big Mom is the type of person who'd sit around and let Kaido take the only person capable of making her giantification dream a reality come true?
Whether it be giants or SMILE, both Big Mom and Kaido need Caesar since he's the only one who can make those possible. That alone is enough to warrant them going after each other if either one tries to kill or otherwise remove Caesar from the other's play. Like I explained in another post, Kaido keeping Caesar to himself is far more important at this stage of the story than acquiring Big Mom's Poneglyph.
You speak as if a Yonkou won't back down from another just because he already has set plans while forgetting Kaido got interrupted by Shanks while he was hellbent on killing WB and Shanks arrived at Marineford scarless