What would happen if a successor could not (or did not) appoint a successor to his teachings, in your view?
Are there any restrictions on who can be appointed successors, or can it be anyone?
Religions brings with them sociological change and normally shifts of powers. Understandingly, this change would serve some parts of the society, and at the same time it represents a threat to the interests of other parts. The problem is that the disadvantaged groups are the ones who are longing for change, while the advantaged and empowered groups find this change against their interests.
That's why normally the mass of the people who initially accept a new religion are the disadvantaged people, who may sincerely believe in it and may not. The advantaged people on the other hand tend to resist the new religion initially.
It is very unlike that within the life span of the founder of the new religion that this confrontation can be settled completely with a decisive victory of the new religion's beliefs, as people tend to defend their old beliefs.
That's why the death of the religion's founder is the golden chance for all the enemies of the new religion, whether they are open enemies or hypocrites to strike back against the new religion. And the easy way is end the effects of the new religion is to hijack it, where a competition to take over the position of the founder is expected.
One the other hand, Religions are about claiming to represent the ideal way of living. It's about claims for certainties and not doubts. As most religions are based on the teachings of the existence omniscient and omnipotent god, religions claims for certainty and for calling for the ideal way of living can’t be fulfilled without having someone who is conveying the God's commands to humans.
Nobody can claim to be conveying god's commands unless he is in contact with God. It's not enough to have access to the resources left by the religion founder. As there is no guarantee that teachings of the founders is to be applied to our life today in the same way that they were applied during the life of the religion founder.
That's why the only one who has the right to appoint a successor of the religion founder is god himself. And the teaching that god is omniscient and omnipotent leads to the teaching that God's way is to have someone representing him among humans. Otherwise how can God bring human into accountability without informing them what laws they must follow?
All in all, at any point of time throughout the long human history god's way is to appoint for humans someone who conveys God's command to his creatures.