Almost everything about Kokochu comes from the Secret History of the Mongols, a Mongolian narrative composed for the ruling house within a few decades of these events, and from the Persian compilation made by Rashid al-Din around 1300 at the Ilkhanid court. Neither is a neutral record, and both were made for rulers descended from the man at the centre of the story.
That shapes what the episode is doing. As it stands, the account is a lesson about the limits of religious authority in a Mongol polity. A shaman may declare the will of heaven and may confer legitimacy, and the moment he sets himself between the ruler and his kin he is destroyed, with heaven understood to have permitted it. The empty tent allows the sky to be seen taking him, so that even his removal is not a defeat for the religious order, only for the man.
Steppe religion is usually known through Chinese, Persian or Arabic writers describing nomads from outside, and Kokochu is the harder case, because the Secret History is not foreign. It is Mongolian, it is close to the events, and it uses the language of Eternal Heaven without translating it into anyone else's categories. It is nonetheless a court text with an interest in the outcome, so the caution it needs is different in kind rather than smaller. What the account does establish, and what later evidence supports, is that named religious specialists operated at the political centre of the early Mongol state, which is not how steppe practice is usually imagined.