It’s really up to the player. Barret is the big ‘emoter’ in the aftermath scene (and in the scenes wherein they come up throughout the story) and you the player can agree with him as much (or as little) as you like. In that scene, going by the text only, Cloud shifts the focus away from the disaster, towards finding the kidnapped Aeris, and to unravelling the mystery of the Ancients. It doesn’t mean he
doesn’t care, but as to whether or not he
does, that’s largely up to the role-player.
....at least until the Remake comes out