Ahem, they were. What you are positing is exactly what happened. Most if not all prehistoric wild horses were ponie-sized. When they were first domesticated in Central Asia they were too small to carry a man so they pulled war chariots instead and that's how they entered the Middle East. Then, further selection made them strong enough to be mounted.
Anyway, even if horses were extinct Middle Easterners would have donkeys to replace them. Up to the 19th century there were some decent breeds of running donkeys, too.
The What-If is not that horses become extinct but that they are too small to ride.
As you (and other) state small horses have lots of uses including pulling war chariots. If civilizations have chariots and wagons do they feel compelled in to find some animal to ride? I wouldn't think so.
Basically I meant the question to be what if civilizations had chariots but not cavalry.