I too have argued that horsemen should not come until long after chariots, as is historically correct, but people wont believe that humans never thought to ride animals until later. I blame Hollywood
The fundamental argument is this:
Person A: "There is no documented evidence showing men rode horses before they built chariots and hitched them up. Therefore, they must not have."
Person B: "There is no documented evidence that shows men did not ride horses before they built chariots and hitched them up. There also is no documented evidence that shows that they didn't. Therefore they must have because there naturally wouldn't be anything that would remain behind as evidence of this sort of bareback riding behavior and obviously it makes more sense to try to ride a horse than build something to make them useful to us."
Person A: "Surely early man was too timid to ride a horse."
Person B: "What makes you think that they would be timid?"
Person A: "Because they feared nature."
Person B: "Why would they have feared nature if they were a part of it? Would it not have taken massive everyday bravery to simply survive?"
Person A: "There's a difference between doing what you have to and guessing that something might be beneficial and taking a massive risk on something you don't think you NEED to do."
Person B: "Doesn't the benefits of horseback riding seem absolutely obvious enough to have at least had some brave souls dedicate themselves to the effort?"
Person A: "No"
Person B: "So one person lacks imagination and that means everyone would? How the hell did we ever evolve into what we were today if we weren't at least a little foolhardy and took some chances on things we thought might give us an edge?"
Person A: "Sorry... just don't see it. Early man had to have been a wimp or he wouldn't have survived. How could we have made it this far without caution?"
Person B: "As we observe modern man, would you say that at the core of what we are that we are 'cautious' by nature? I don't think thousands of years has taught us to be risk takers - we would've not emerged from prehistory at all if we weren't."
Person A: "Ok, point taken but there's no evidence that man ever rode horses before they built chariots."
Rinse and Repeat.
Ok, a little deeper. Let's take a look at the 'evidence' that we did not use horses before chariots.
1) There is no wear on the teeth of the equine skeletons that date before the chariot. This means we did not use a bit and bridle and therefore could not have ridden and wielded a weapon at the same time.
Refute: So we didn't invent the bit and bridle before the chariot. Fair enough. But don't horses have manes and are adeptly ridden without reigns by riders with a distinct relationship to the animal? Wasn't early man more in touch with animals than we are today much as native and indigenous people are? Wouldn't that same attitude of being part of nature rather than apart from it have led the earliest riders into a relationship based riding experience between the mount and rider? Would not this 'relationship' rather than master and beast style riding have enabled the far more physically adept early man to do what we would consider ourselves impossible today?
2) There is no way we could have forced the animal into danger (combat) without having the control that the reigns would provide.
Refute: Again, relationship. Horses are actually pretty intelligent and emotional creatures. Fear is not their only emotion. It isn't the only motivator for us. Science has recently objectively proven that dogs do actually care about their masters, not just what their masters can do for them. Why would it be any different for any other mammal? The horse could be inspired to support the will of their companion human out of emotional bonds quite easily I believe, and throughout history, it is that very bond of companionship that has enabled even humans to psychologically dispute their natural urge to flee from battle. I'm sure any practiced rider will be able to go into much more detail on this, relating from their own experiences what their mount is willing to do, and endure, when there is a true bond of caring between them. So point #2 must truly stem from a modern viewpoint that animals are purely basal and incapable of the complexity of feelings that humans posses.
3) We have not found equine skeletons on the fields of early battles where we found human skeletons.
Refute: Most indigenous people have a deep respect for what they kill and preach that one should use as much of the kill as you can, all of it if you're capable. While dead humans may have been treated much differently, the sense of responsibility towards the fallen equines may well have inspired a behavior to remove the dead from the field. Besides that, the horse was, as has been pointed out, quite an acceptable food source. Why would the victors not take part in the bounty of the spoils? Again... another cause to remove the animals from the battlefield.
4) There is no written record of horses being used in battle.
Refute: We're talking about a place in history before the written word. Writing came after the chariot was invented so... of course there wouldn't be.
5) There is no artistic record of horses being used in battle.
Refute: It very likely was quite uncommon. Horses were probably too valuable to use in this manner very often. Early man probably knew HOW to take them to battle but were reluctant to for the same bonded reasons that the horse would be willing to if they did go to war.
However, I think we can wager that horses WERE used in numerous conflicts. Scouting, patrolling, hunting. That it was not only possible, but likely due to the obvious benefits riding on horseback would provide - even if the horse were more often simply used to get from point a to point b faster and with less effort on behalf of the rider. It would not, however, have been until civilized man's idea that we are here to conquer the natural world and bend it to our will as opposed to the indigenous attitude that we are simply a part of nature, that we would begin to see truly militaristic concepts emerging that would add the horse as a solid part of the battlefield.
The Chariot rapidly emerged as a part of this blossoming into what we call civilization. The Sumerians were clever. We have a lot more mysteries to resolve there about how they raced through so many technological discoveries practically overnight. And maybe THEY didn't use the horse in warfare until this sudden emergence of civilization because the horse was far too valuable to risk for that in the fertile crescent. Particularly after they began using the help of the horse in farm labors. But does that mean that those tribes that existed without written record for thousands of years before and after this point in other places of the world where horses also existed would not?
Let's also make another counterpoint here and this is that WARFARE itself is rarely depicted in early art and yet we find much more evidence for it throughout the prehistoric record.