Sumerians had carts and wagons pulled by onagers, but no war chariots. I doubt that they would ever come up with that idea, for the reasons Revachah has pointed out. War chariots came to that region from the North. Without horses in the steppe, it won`t come from the North for a long while, and I wonder whether it will come from anywhere else.
I agree with Revachah that horse riding came from horse hunting. Doesn`t mean, though, as Workable Goblin pointed out, that, once either donkeys or camels reach the Eurasian steppes, they won`t be ridden. As you pointed out yourself, riding animals occurred in many places separately from horse culture. That`s not the same as assuming that everything we did that was related to the horse would have come about in an ATL only with other animals, here I totally agree with you.
Riding donkeys is a completely different experience from riding horses. It won`t lead to the same cultural and technological results that riding horses did, even though that`s biologically not impossible. But it will change things considerably nonetheless.
I believe in creative soldiers, too, but I still think asinary is not going to happen, while donkey-chariot warfare might, and so might elephant-chariot warfare, and donkey-mounted infantry. Problem is, it´s not going to develop in places where donkeys aren`t ridden usually: If you`re stuck between a rock and a hard place on a battlefield and you can`t flee with your donkey-chariot, you`d much rather flee by foot than by riding your donkey because your donkey isn`t bred for speed, it´s bred for strength and endurance. Now, if steppe peoples begin riding their donkeys because they want the oversight and they need to move around a lot whilst not having to cart around stuff all the time, that`s a different piece of cake because they might breed their donkeys for (more) speed (albeit maybe not the speed of horses since that whole idea would not be in their cultural imagination). Using them in mounted warfare is still not going to become a big thing, but at least these donkeys wouldn`t be super-slow.