Wut?
About the main point, ofc, Zoro won't fight Kaido, nor he will kill him.
Kaido was talking about Straw Hat Luffy and Kinemon praised him, no shit about Zoro. Zoro didn't exist in that conversation.
Kinemon as the leader of scabbards talked to Luffy, the leader of the alliance before he went to the roof.
When Ashura saw Luffy's CoC, he remember Oden, his leader. CoC users are the main characters, the leader, where everyone put their faith, etc.
Zoro has no attachment/relationship/ties/motives with the boss. Enma? It was a mere hype to boost his strength cause Sanji had his own powerup before him and he wouldn't stay behind obviously. Everyone cut Kaido and Oden cut Kaido with his both swords. So the crap about Oden's will through Enma to Zoro is crap and the same with "the only sword which cut him", that's all.
People were saying before that Zoro would cut Orochi and with this one he has more "relationship/ties/motives", and now he is dead lol.
Zoro is not a main character in Wano. The story does not spin around him.
But it does around Luffy.
As for a possible clash with BM, well, maybe. After all he is running around the place now with Apoo so he may arrive where BM and shot some attacks at her.