I think that if Cait Sith had died then and there, and never came back it would have defeated the point of Shinra being able to follow them/know where they were going next, it'd also leave it unexplained. Cait Sith's presence was there to link Clouds party and Shinra together as a team regardless of them being enemies. Because if you think about it, once they were out of Midgar, they had nothing to do with Shinra other than being on the same mission to find Sephiroth. Cait Sith connected them, giving us a brilliant storyline in my opinion. He may have been a "doll" or a big stuffed moogle with a cat on top but his importance in the game is noticable. Besides, he's cute