Chingiskhan didnt want to conquer Kharazm originally, instead he offered trade relation, furthermore, although furious, he didnt immediately send an army to deal with Kharazm after Mongol trade delegation was murdered like perhaps most would assume him and Mongols in general would have given their later practice.

he sent special envoys to investigate and demand hand over of the guilty. which was an action fairly proper and rightful. But, the Kharazm didnt respond correctly and showed further arrogance and insults to Chingis  which resulted in conquest of Kharazm.

Batu's creation of Golden horde was no doubt extension of the Mongol empire as a whole, but specificly speaking, before Batu embark on conquest of Russian and raids in eastern Europe, Shuchi and Chagetai already were showing signs of disunity as they fight each other over conquered cities and lands for themselves.

Batu as heir of Shuchi's khanate moved to far Russian-European territory means abandoning claims in CA to other princes, which serves interest of both.

For Chingis heirs, they are creating a vast empire never before seen, but also at the same time splitting themselves, and that split is certainly facilitated by the fact that there were diverse and distinct cultures where the Mongols ruled.