The thing is, they didn't have a shared goal. I mean, Big Mom surely wanted to be "PK" and create her utopia where all the races of the world could live together, but I don't think that she would be able to defeat her final roadblock, which was Kaido (I feel that she would probably go down earlier, either against another Yonko, an admiral or even Loki). IMO, Kaido was pretty clearly the "final boss" of their Alliance, so they would probably stick to his grand objective in the end: killing himself in a massive war.
His dream was actually to be defeated by Joy Boy and it was a destined thing, predicted even by Oden, so he would probably speed up his end after Luffy's failure. Maybe they would take down Blackbeard and take back Hachinosu (never seemed to be a thing for them, so idk), maybe Shanks would go down as well... they could even try to subdue Elbaf (and much likely fail; due to their bad blood with BM they'd probably prefer to be destroyed), but ultimately they would lose against the WG.