@Kaarme Well, what you said does makes sense and it's probably what should happen. But then again, even in today society where free thinking is a widespread concept a corporate or a big house/clan that have their previous leader as the one that holds power instead of the current leader (a.k.a Puppet Leader).
So in my opinion as to why this happen:
1. The old master still holds the power and the transition of power is either took too long to do or it is controlled by the old master how and for how long it will be done.
2. The old master still holds the secret of the family and just handling the leadership doesn't makes him the rightful leader. And even if it does, it will make the house split into 2, those who support the new master, and those who still support the old master. This will result in weakening the house and lowering the efficiency when dealing with the demon
3. This is just a plot hole made by the author
Owh, and thanks for telling me that. It never really crossed my mind