i find it quite interesting that you take the direct other part of the thought. I like your idea.
my idea so far was Yamato being the son of Oden and Toki, born after Odens Death. Kaido raised him as his son, so nobody expect it. But Yamato grew as Odens son, similar taste, similar character, but well trained by kaido.
so kaido sees him as his son, but he has the genetics of Oden, and raised him as son, cause of respect to Oden and his strength. And also as a way to get the strength of Oden into his crew.
and that brings us to the big announcement of kaido. Following that thought, Kaido could reveal exactly that and make Yamato the shogun of Wano, as legitimate follow up to Oden. Also he offers Orochi to the scabbards, or even kills him by himself.
that’s my thought so far. Much holes and a lot to think about, but came to it a few minutes ago, so give me time to build something around it.