The Vivre card mentioned that King was with Kaido that time he tried to go to Marineford. I was surprised to read this because I thought it was kind of obvious that he would bring a few people, at least to navigate the ship and stuff. Then I got it.
The Vivre card mentions it because Kaido only brought King, or maybe King decided to come along. They didn’t need a crew because they didn’t sail to Marineford.
I believe that because of how isolated Wano is, Kaido got the news about the upcoming war just a couple of days too late and there was no time to prepare a ship, gather some troops and get going.
But he had to be there because it was his last chance to fight Newgate to the death, and then have a go at the Admirals, Sengoku and the entire Marine HQ. Kaido figured he would fly to Marineford in his dragon form. King, his only commander capable of flight, tagged along leaving Queen in charge of Onigashima.
It’s also mentioned in the Vivre card that Kaido hasn’t fought a worthy opponent since Oden, and we know that Shanks became an emperor 7 years ago. They probably never fought since he got the title, maybe even longer than that.
Therefore, I think they only clashed briefly enough for Kaido to realize that he had met his match. Kaido wasn’t sure if he could defeat the Red Hair Pirates and goes to Marineford, so he decided to come back to Wano. From there on Kaido considers Shanks one of the 5 people capable of defeating him.
This explains why Shanks and his crew got to Marineford looking like a million bucks and faster than expected. The marines expected a huge naval battle that would wear both sides down and go on for a while, instead it was just a quick intense clash.