Kaido wants to start a war, Kaido threatened Big Mom while the Marines were listening, Big Mom showed up and all of a sudden Kaido turns into a panicking 5 year old?
There is clearly more going on here than what it looks like on the outside. Kaido does not want see Big Mom at all otherwise he would have personally gone and blasted the ship. And Big Mom came for Luffy and did not come to start a war with Kaido. And Kaido could have easily given Luffy to Big Mom but he won't for some reason. You're here acting like Kaido is Shanks 2.0 who is trying to keep the peace but no he's an alcoholic psychopath with a possible personality disorder. Big Mom has something on Kaido that we still don't know yet.