Avi, you may have a point about what sometimes happens when there is no living rebbe, but in practice, there have always been several Breslov manhigim in each generation who functioned as leaders, giving people advice and so on. For example, today there are many chashuv Breslov leaders today–Rav Shalom Arush, Rav Elazar Kenig, etc.
Rebbe Nachman was opposed to hereditary rebbes, which most other chassidic groups have had, perhaps because it tends to create a social system in which people rely vicariously on the rebbe’s spiritual attainments instead of focusing on developing themselves by implementing the rebbe’s teachings. For example, as recounted in Reb Noson’s biography, a chassidic rebbe confided in Reb Noson that he was jealous of Breslov chassidim because his chassidim mainly came to him for blessings about material problems, while Breslovers were focused on enhancing their avodas Hashem.