Hmm... I thought the game made pretty clear that Blitzball was a very important sport in Spira, that it gave the people something else to worry about than Sin and that the players 'carry huge responsibility'. Not that they didn't care about saving the world from Sin.
Besides, considering how quickly everything went ("when we arrive, the tournament will start immediatly"), the whole tournament probably didn't take more than a couple hours, considering it's 6 matches of 10 minutes... I really don't think it was that big of a deal.