King and Jack are not scared to face the flying six at all. I mean, why should they?
They don´t have any other choice at this point.
As we can see Kaido crew is build up with monsters and a system where the strongest gain the titel/command.
Kaido as someone who face anymore don´t fear anything too, I mean Jack attack a ship with a Admiral/FleetAdmiral+other VA in blank point without fearing them.
That is how Kaido crew work, no fear but with all monsters who clash and battle each other.
It could even help them a lot in other encounters, fighting for the titel/rank means to don´t let your guard off for any time, im sure until now King/Queen and Jack training and training knowing that their are other monsters who want their position.
In compare the Big mom pirates are family more, someone as Perospero could control Smoothie even though he had less power.
Also they don´t fear to lose their position because of their family member, they wouldn´t fight each other.
Probably in same way better for a team cohesion but less effectively as the beast pirates who have more experince and training via crew battles.
It make sense why Kaido want to show Linlin his new troup, he trust them a lot and also he seems proud of the flying six, I think he see them in same way as his calamities. But something which I kinda find nice, Kaido ask first his right hands if they agree with the battles, with kinda show that Kaido little has respect to his calamities.