I don't think Kaido was interested in killing Whitebeard specifically but rather wanted to join in an escalate the situation so he could get the "glorious battle" he was looking to die in.
Unfortunately he received none of that and thanks to Oda's dipshit grandmother and his obsession with cramming together flashbacks we had an overarching villain defeated with shit writing and so unceremoniously that people aren't even sure he's dead.