So Kaido definitely knows something about D's. First we had his comment about smiling/laughing which was cut off, and now we have him commenting on how without proof of Luffy's death, people will continue to have faith in him.
It makes sense considering he served under someone with a D in their name, and that person was capable of bringing together all these pirates who would otherwise want nothing to do with each other.
I'm thinking he must have really looked up to Xebec when he was younger, and possibly even witnessed his death first hand.
Now I'm wondering if Xebec is the one who had Kaido's DF before him, and Big Mom was somehow able to take the soul of the Devil Fruit after he died, and give it to Kaido.