It’s a good question! Since I’ve never worked directly with ambassador animals, I reached out to a couple contacts who work in program animal departments at accredited zoos and the president of the Feline Conservation Federation.
Cheetah are very behaviorally different from all of the other big cats. They’re designed to out-sprint their food, not ambush it. This means two things: they’re less likely to pounce on interesting things (while not outside of their behavioral repertoire, cheetahs generally hunt by swatting the legs out from under their prey), and their survival strategy / response to stress is to flee rather than fight. While they’re still strong wild predators, these behavioral tendencies make them a lot less dangerous to work with in a professional free-contact capacity than tigers and lions.
In addition, a major reason zoos will do public walks with cheetahs over other big cats is that a full grown cheetah can be physically stopped by two able-bodied adults. That’s why you always see cheetahs in public with at least two handlers holding separate leashes, and even then there’s almost always a trained backup present. There’s no similar possibility for using the weight of humans to physically control an emerging situation when working with other big cat species. (As an added note, since cheetah claws don’t retract, they’re naturally dulled by running – they’re still plenty injurious, but not the primary weapon they are on other species of felid).
And lastly, cheetahs are highly tractable. I don’t think we know exactly why they do so well – I’ve seen it attributed to their timid tendencies, but that’s not the whole story – but cheetahs that are hand-raised well frequently become stellar ambassador animals. The addition of companion dogs improves even further upon their calm demeanor, but isn’t necessary for a well-raised and trained cheetah to be calm and confident in new situations and around the public in situations when no other adult felid would be an appropriate choice. (Young clouded leopards are often ambassador animals as well, since the species is most successful in human care when hand-raised, but few zoos continue doing programs with adults).
How zoos and outreach organizations are able to bring cheetahs and clouded leopards in public is basically up to the discretion of the facility’s USDA inspector. The animal welfare act regulates the specs of their permanent home, but what public forays require are decided on a case-by-case by case basis with each facility. Generally, there is at least one physical barrier of some sort between the public and the cats at all times – but sometimes the public is simply kept at a safe difference, such as during presentations that take place on a stage or in an amphitheater. It’s worth noting that even the much more stringent standards for felids proposed in the Big Cat Public Safety Act (which I flatly do not support because it’s a misleading, biased, and badly written piece of legislation) contain exemptions that allow facilities to continue using cheetahs and clouded leopards as ambassador animals.
Compare what cheetahs normally eat to what other large cats normally eat. Cheetahs mostly go after small prey, smaller than humans. When they catch their prey, they swat its legs out and bite its throat, they aren’t designed to rip at prey like lions do. I don’t think a cheetah could accidentally kill a person unless it was trying to playfight and the situation was allowed to go much, much too far. And I have no way to confirm this, but I imagine a cheetah, being a smaller, weaker, more timid animal, would be much easier to fight off if it did actually attack a person and was in a situation with no one else to fend it off.
Basically, they’re smaller, they’re more timid, they’re easily trainable, they mostly hunt smaller food, and they aren’t aggressive.

