Not all the same things happen to all Avatars. If that is the case, the Fire Nation could have easily studied Roku's life and used it as a blueprint to capture Aang. All Avatars have in common are their ability to bend the four elements and use energybending.
Actually, it is speculated that since Avatar Kuruk lived a careless life without any stress, the Avatar Spirit was reincarnated to Avatar Kyoshi, who lived a serious, more controlled life, to somewhat balance out what Kuruk did in his past life.
Because of the approaching war with the Fire Nation, it was decided by the monks that Aang was to be told that he was the Avatar at a young age, so that he could start his training early. As he was told at such a young age, the pressure was too much for him, and he ran away. This has not been the case with any other known Avatar.