Yserbius, no, a Rebbe does not leave his responsibilities to someone else. It says “Aseh L’CHA Rav”, choose one for yourself, not that the Rebbe is chosen for you.
A Rebbe is who he is because people choose to follow him. If people choose to follow his son, that’s their decision. In the early days of chassidus, most Rebbes did not have their sons take over. It’s only become an issue more recently, when chassidic groups control vast amounts of money, that the sons would insist on taking over.
Even R’ Yoilish didn’t want to become Rebbe at first, but people insisted he do so.
