There’s a number of sports that centaurs would be worse at or outright incapable of. Gymnastics being an obvious one. I’m not sure about swimming; how good are horses at swimming compared to humans?
Also polo might be popular, or a modified version of field hockey.
Given horses vulnerabilities to a number of cattle diseases, then there would likely be very few centaurs. The relatively small number centaurs would have been no match for the larger number of human being and they would have rendered extinct as a result.
There probably wouldn’t have centaurs in the New World as there were no horses in the New World (at least in large numbers) until after Columbus arrived (the Vikings may have brought horses with them to Vinland but they obviously took them with them when they were expelled by the Natives or they died after being abandoned). That would have left centaurs in the Old World and even then in a limited area (horses weren’t common in Sub-Saharan Africa until the 11th century).
Finally, the limitations of their anatomy wouldn’t have helped. They can’t turn around without turning their entire bodies meaning that they could be snuck up on from the rear quite easily. They can’t climb trees, so they couldn’t avoid a number of predators. Nor could they easily navigate rough terrain, meaning that they would be limited to relatively flat ground, again leaving them vulnerable to predators.
Centaurs would be a sad animal if it existed that would have a limited lifespan and which could easily be controlled by human beings who have fewer of their limitations, while still retaining their intelligence.
Horses get by in most of those limitations by living in herds. They also navigate pretty darn well on rough terrain.
Would their be centaur little people? With pony or donkey bodies?
Actually, horses do poorly on rough terrain. Their hooves get stone bruised and split and they can’t easily navigate steep hills or in the mountains. Even out West where I’m from (hence the name) you only see wild horses on relatively level ground. And horses can see behind and beside them quite easily (look at the setting of their eyes). Human eyes are set forward as we are predators; but we can also turn around quickly to discern danger behind us. Centaurs would lack this ability.
Ponies and donkeys are slightly different animals from horses. Same general family; somewhat different animals. Ponies continue to exist as a human creation due to breeding and they would survive poorly in any environment with a number of predators (they evolved as horses which matured living in marginally habitable areas.)
As far “cent-donkeys”, these would even be more vulnerable due to their smaller size and slower gait it would seem.
Eh, our horses were pastured on hillsides at times. But then, they’d always have those legs on the one side shorter than the other. 
What would centaurs think of humans using horses to ride or as draft animals? Would they view humans as oppressors of their cousins? Or would they think of horses more like the way we think of chimpanzees and monkeys?
Depends on if they’re “undocumented” centaurs in which the top half is Mexican.
Why do you presume they’d have the same vulnerabilities as horses? They aren’t horses - they are half human by appearance, and who-knows-what genetically.
Yes, according to Disney, and they didn’t wear bras either.
The original version of Fantasia (1940) featured bare-breasted female centaurs, until a later Disney-correct version was released which sanitized the scene by putting bikini tops on them.
Maybe they’re like giraffes with really buff necks?
I can just see it; a centaur among a herd of horses saying to one of the horses “I don’t know what the heck it is; it just started as a bump on my neck and grew.”
Only if they were morons. There’d probably be some organisation called CETH or whatever, but the rest of the centaurs would be smart enough to not fall for that kind of stupidity.
Makes sense, I guess. Why should only humans get morons?
While I agree about the Americas, I see no reason why Africa couldn’t have been inhabited by a centaur-like creature with the hindquarters of a zebra.
Yeah, that Disney Fantasia segment is funny. They have not-really-all-that-racist black woman-zebra centaurs. Plus a totally-racist gollywog-donkey centaur servant.
Centaur porn is already a thing in our world. Can you imagine the centaur porn everywhere if centaurs were real?
I wouldn’t want to live next door to one of those damned grass-chokers.
While zebras are certainly “sturdier” than horses, the inability to see behind you would mean that centaurs would be vulnerable to animal predation in a manner unlike that of other animals. The animals in Africa survive because they are herds and have evolved the ability to avoid predators.
I’m thinking a pony-sized equine part, for Centaurs.
Easier to go indoors, & use elevators.
It would be less of a problem if centaurs were, themselves, predators.
After all, humans can’t see behind themselves, either. Nonetheless, we’ve done pretty well. We’re omnivorous and capable of being predators even if some of our closer relatives are vegetarian (gorillas and orangs, we all know now chimps will also kill and eat meat).
Why not a tool-using centaur that is also an omnivore and possible predator like humans?