Most of the hunter gatherer populations on this planet adapted to their environment and used what was around them to succeed. Mongolia doesnt really have a lot of vegetation, like trees and bushes, so using horses and strong bows to cover the distance quickly and have a weapon with some range was probably born out of necessity.
Likewise breeding good animals is part of our dna, it also explains arranged marriages in some respects, but viking burial sites have also found Norwegian Elkhound remains buried with them.[1] Even the ancient Egyptians used to have their pets buried with them which included things like gazelles and baboons![2]
The lifestyle back then also meant kids would have learnt a trade from a very young age so expertise would and should be alot higher than today when thinking about the more wide ranging subject matter that is taught today, this perhaps explains why High Functioning Autistics are the experts in their domain of knowledge because they push things to the limit and know everything inside and out, making them ideal employees for some jobs.
[1] https://ahcfargo.com/client-resources/breed-info/norwegian-e....
[2] https://www.nationalgeographic.com/science/article/news-egyp....
The source is one historian from Kazakhstan, and his main reasoning is that the common words used have meaning in the Kazakh language. A language that seems to trace it's origins to The Golden Horde centuries after the Mongols.
Beyond that, most of the Eurasian Steppe had a strong history of using horse archers for millennia. That wasn't unique to the region that is now Kazakhstan.
Maybe there's some more compelling argument, but the one you presented seems weak. It's possible the Kazakh language bits had some meaning lost when translated to English, as the way they are presented is somewhat embarrassing. It somewhat sounds like he's saying modern people from Kazakhstan are more closely related to the leaders of the Mongol empire, but that's a very different claim than "Genghis Khan a Kazakh."
[1] Genghis Khan a Kazakh?
https://www.upi.com/Science_News/2004/04/06/Genghis-Khan-a-K...
[2] A horse in a burial-Memorial ceremony of ancient nomads in Kazakh Altay region:
https://www.researchgate.net/publication/286946217_A_horse_i...
[3]Horses and Humans: Ancient Bond:
https://astanatimes.com/2023/04/horses-and-humans-ancient-bo...