All of which is several generations of Great Khans after Genghis and irrelevant to the thread topic. The Khanate managed on in a fairly stable climate until about Kublai Khan and the three Khanates in the Khagan began to develop a more independent power structure by the 14th century. The Illkhanate adopting Islam, the Chinese territories establishing the Yuan dynasty (which would reign for about a modest hundred years), and the Golden Horde doing whatever the hell it wanted. But prior to the death of Kublai and even after the death of Genghis Khan these groups within the Empire were integrated units of the larger Mongolian politic.

The succession from Genghis to his sons, and his sons to their sons weren't anarchy as you're trying to describe. And the entities of the greater Empire as described were all the same elector powers for the appointment of future Great Khans.