Why don't we ride zebras?
I will answer your question by answering a different, but related, one: Why don't we ride zebras?
The zebras are further away from modern horses than the Przewalski's horse, but is still a species under the Equus genus. They look like horses and can carry up to 350 kg weight, so why don't we ride them? Short answer: they can't (easily) be domesticated.
Requirements for domestication:
In order use an animal as a common mount, it needs to be domesticated and then tamed1; in order to be domesticated, it needs to have the following traits:
1. Will breed in captivity. This is highly crucial, if you can't breed them in captivity, then you can't influence the traits of the species since you can't get the two nice and calm horses (or whichever animal you want to domesticate) to produce offspring.
2. Fast growth rate. They need to reach adult size within a reasonable time, such as less than 5 years, in order to be as useful as possible. If they take long time to reach maturity, then we will get less food and work output for the time and energy invested in the livestock. It is also a requirement for proper breeding as you will want the animals to have more generations born per human generation so you can selectively breed those animals with the best traits quickly.
3. Proper diet. We need to be able to feed it with what we have, so a picky eater will be problematic to feed. Horses, cows, pigs, dogs and other domesticated animals are not particularly picky. Horses and cows eat grass, silage, grains, while pigs and dogs eat almost anything.
4. Manageable size. Very large animals require a lot of food and space, which means that there would be a lot of time spent getting food for the animal as well for the human families. For grass eaters, all you really need is a large enclosed field where they can graze. However, it will become cumbersome if they require too much food, especially in colder regions as you need to prepare and store food for the winter. Meat eaters need to be relatively small to consume reasonable quantities; a lion, for example, requires 5-7 kg meat every day, which is not that insanely much, but it means that you would need to keep a lot of other animals to use as food for the lion in the long run. Dogs, on the other hand, require generally less than 0.06 kg raw meat/kg 2 per day and can eat other things than meat.
5. Good temperament. One needs to be able to trust a domesticated animal to a certain degree. If they are likely to become angry and attack for no reason or if they are likely to panic, then they are difficult to handle and difficult to domesticate. Horses and deer are prey which flee at the sight of danger, which is a trait which needs to be removed by selective breeding (tamed horses still have a tendency to panic, but we have largely managed to calm them down as a species). Lions and tigers are large enough to potentially consider humans food and therefore pose a threat.
6. Social hierarchy. This is not a hard requirement, but if the animals have a social hierarchy, then they will follow a leader. If a human become the pack leader, then they will follow the human.
So, how does zebras fulfill these requirements?
They have all points, except number 5, because zebras are stubborn, mean bastards. They are highly unpredictable and while we have attempted to domesticate them, they have been too difficult to work with. Another reason is that, aside from their looks, they have no qualifications that outperform a horse so no one had bothered with the effort of truly taming and breeding away all the nastiness in the zebra. It has simply been too much of an effort when we already have horses.
How does all of this relate to Przewalski's horses?
The Przewalski's horse is the nowadays only living true wild horse, it is currently red-listed as endangered since it was extinct from the wild, but later reintroduced from captivity. It is related to the domesticated horse but have, as it is a true wild horse, never been domesticated and mitochondrial DNA analysis shows that it is not the ancestor to our domesticated horses. However, being closely related to the domesticated horse, the behaviour described is that of horses. Modern horses comes in various looks, sizes, temperament, and other qualities, all which have been sprung from selective breeding throughout history. It is, therefore, highly likely that had we domesticated Przewalski's horse instead of the Equus ferus caballus, then we would have essentially the same type of horse as we have now, but with some difference in looks and sizes. Probably the same with zebras, if humans had taken the time to breed away their nasty traits. Although, zebras are more related to asses than horses, so the difference in looks would be greater.
It would not have made any difference.
1: You can, of course tame individuals without domesticating the entire species, but it is more difficult to tame a wild animal and, thus, it will not be a common mount.
2: The Calorie content of raw meat varies depending on where it is from; I used 2000 kcal/kg as average based on this. I also tested various sizes of dogs and noted that it was not a linear relationship, so I used a 35 kg dog which would be equivalent of an average German Shepherd. I used "Working dog (heavy duty)" for activity level to get something that would simulate the maximum food requirement of the first domesticated dogs. The total would be less than 2 kg raw meat for a 35 kg dog that works very hard every day.