The thing here is the axe has legitimacy this generation and the next, most likely. If we unite the Thunder Horse under a favorable heir, the Dead Priests can offer any other heir the axe to immediately produce a rival strong claimant, which means the Thunder Horse will either unify under the axe(bad for us), or go into succession war(good for the DPs).
Because the nomads are not a unified bloc. The cart currently has a "significant" following despite the rider being far from favored in succession. Throw the axe in on someone with a strong claim and they'd back that guy to a large extent.
Remember, this is exactly the sort of deal they offer to their client states.