Yeah, maybe. But catching animals is always part of the herding, usually not daily business, but in many situations, least not of all catching for the kill. Horses are constant grazers and not as feed efficient as cows (or goats and sheep) this also means that they cover a lot of ground a day and they are skittish flight animals. So you can not compare them to cow/sheet/goat. Also horses haven been domesticated quite late in history, compared to the other animals. Botai horses clearly were ridden, so riding horses was part of the horse herding business and was developed in parallel.

The Botai culture was believed to be pasturing horses but initially without proof of horse riding, therefore the idea that before riding was developed, the horses were bred for meat. As far as I understood this is kind of debunked now, as there were finds of horse skulls from that time with marks on their teeth originating from snaffle bits. To me the idea of herding an animal good in running like horses as a walking human seems ridiculous.