Main issue on politicization is timing.
Yes, it's inevitable, but a shaman base in the center of politics, before they developed their own systems and techniques to manage shamans, would lead to taking strong influence from the extant favor trading.
You can contrast with RL religions:
-Judaism, Christianity and Islam rose to prominence, with their Great Temples being based in their political centers. In times of chaos, the religious system was basically the secular state system as a whole, and acted to worsen fracture lines and seize power of their own.
-Hellenism(can't recall the proper name for the Greek system), Buddhism, Taoism, Hinduism and Shinto rose to prominence with their centers of religion at a prominent, but not political location. This led to them being focused more on their rites and the people early on, and then later they actively resisted political influence much more strongly, by having their own spiritually organized hierarchies which were difficult to intervene in. This meant that while individual temples or sects may influence individual kings and nobles, the overall religion largely stood apart from the administration save to stabilize things.
Timing matters. If the Governors seized power back in ChariotSon's crisis, we would have a far less coherent Oligrachy than we do now. Political changes will happen, but changes which happen slowly tend to serve the purpose of their social caste better than swift changes.