Article is 2018. I believe this is a pretty old article and original horses are now know to have come west from the Americas to Asia via the bering strait, but am not able to look it up at this time.
The article is about domestic horses, not about their extinct ancestors. The article says that previously the Mongolian breed were believed to be the last truly wild, undomesticated horses (predating domestication 5.5k years ago). Now they believe that they in fact are descendants of domestic horses. Making all living horses descendants of domesticated horses. But yes, the wild horses the Botai domesticated evolved in the Americas.
That is a link to a Wikipedia article. The article references an ancestor of the horse migrating over and back and forth from North America to Asia, but not the horse itself, and certainly not domesticated horses, which originate in Middle East / Asia.
Somehow? Before steam engines, one needed another horse, loads of persistence, or large traps, to catch up with just about any horse.