I think the issue was developing a bit to control/domesticate the horses, which some archaeologists found evidence of about 4800 BC in the Russian steppes. The Yamnaya culture, referenced in the article, spread from present day Georgia/Azerbaijan, reaching the Danube Valley and the Carpathian Mountains in about 3100 BC. My source: an excellent book called “The Horse, the Wheel and Language” by David W. Anthony.
I seem to recall that the US plains indians did quite well without using a bit or even a saddle. The development of the stirrup was also an important development for warfare.