Better late than never: This BBC article highlights claims of 9k year old domestic horses in Saudi Arabia.
When I teach about animal domestication, I make a distinction between different types of domesticates:
commensals: These are mice and pigeons, but also pigs, guinea pigs, cats, and other, more acceptable, animal companions. These animals are domesticated largely through their attraction to human-created environments.
herd animals: sheep, goats, cattle, etc. These animals were domesticated from hunted populations, largely through substituting humans for the leadership positions within the herd structure. The original purpose of domestication was for the primary products (meat, blood), but secondary products (wool, milk, traction) could become important later.
transport animals: elephants, horses, llamas, camels, etc. These animals are often difficult domesticates, in that their social structure isn't as easily dominated by humans, or they cannot be allowed to stay in their normal wild social organization, because the males will fight, etc.
I usually tell my students that most transport animals were domesticated far later than the other kinds of domesticates, because they are more difficult to control, and because the domestication of animals for their secondary products alone (transportation, traction) is seen as unlikely before fully domestic economies had already formed.
This article on early horse domestication suggests that horses were originally domesticated for the meat, not the transport. If that's the case, it's not as surprising to find horses were domesticated just as early as sheep, goats, and cattle. Horses fill a different ecological niche than those other herd animals, and may have been a useful way to exploit some parts of the desert environment.