Also, how did Micheal's remaining on the island (until he had no other choice than to leave) further his supposed objective of finding out who was the traitor?
Because he didn't know it for sure until Fredo made his fatal gaffe at the Superman show. Then he immediately dispatched Busetta to kill Roth.
I didn't say that he stuck around for the presidential party specifically because he needed an alibi. He probably stuck around because he and Fredo were squiring around that high powered delegation of US politicos. Recall that, at the party, some of the people were asking after Roth. They'd be asking after Michael if he, too, failed to show--and might have put two and two together after Roth turned up dead.
That leaves the question of why he invited the politicos to Cuba. Guess: Another way to "relax" Roth by proving to Roth that he was cementing his political flank as Roth's designated successor in Cuba. Another guess: A way to prove to Batista and/or his successors that he had enough clout to bring high-powered politicos to the island. And, possibly, to establish an alibi and use political clout to shield him.
