Leader

You may argue the story of Li Bai was an exception, and it is entirely logical to choose the best expert of a field to be the leader.

Alas, I'm afraid you read too much Jin Yong (金庸), and are obsessed with the romantic idea that leaders of all martial sects must be the best swordsman of the group; if a leader leaves - dies, steps down, or steps down in the passive voice - the successor of the post should be chosen by a fighting contest. As along as he fights well, he would suddenly becomes an outstanding leader once he takes up the post - like Mount Heng Sect (恆山派) under Linghu Chong (令狐冲).

Maybe Ming Cult (明教) under Zhang Wuji (張無忌) is more close to reality.

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As to Li Bai, there are rumors saying that Emperor Xuanzong did want to make him the prime minister, but the idea went in vain because it was strongly objected by Gao Lishi (高力士) - whom was badly offended by Li Bai when they first met in Chang'an.

When I read more about the life and career of Gao from The Book of Tang (唐書), I am convinced the powerful eunuch did make an objective advise to the Emperor.