I mean didn’t he say he wanted revenge on Kaido and to dethrone the worlds strongest creature?
He just said he takes Kaidos head, when they went in direction onigashima. Before, on the table of the alliance with Apoo etc., he really didn’t cared for which yonko they go.
And on the way to onigashima, he just picked the Yonko which he knew is there. At this point, there was no clue of big mum being and additional enemy there.
At the end, he never wanted something else from Kaido as the title and the RP, he never declared a personal grudge.