This would keep up the population but they could become domesticated like common house cats and the world would loose a fierce predator and a key stone species in the wild. But we could solve these problems by domesticating some of the wild cats and keeping some in the wild. But they are in fact wild animals and they could also be predicted to act wildly. This might not be a problem because normally most dogs have the ability to become rabid or wild but also can be one of the most favorite pets, this could happen with cheetahs.
One of the best ways to promote environmental health and diversity, as well as protect threatened and endangered species, is to legalize and regulate the hunting/trading of them. There will always be people who want exotic pets. It's better for them to go through channels that keep them in touch with the status of the animals in the wild and have their money go to protecting and proliferating the animals than avoiding the law and padding the pockets of poachers (Even though a lot of poachers are feeding and protecting starving African and Indian villages, so...) The "More tigers in captivity than in the wild" is not a bad thing, either, as long as the tigers are thriving in captivity. The problem is that Cheetahs are dwindling in captivity, so they're going extinct on two fronts. Even in a worst-case scenario of tigers going extinct in the wild, it won't be a death knell for the species as a whole.
Cheetahs are still dangerous wild animals and this could have deadly consequences. Not only could it have deadly consequences for Americans to keep cheetahs as pets, but there is no guarantee that the Cheetas will breed in captivity and that the population will be increased. Instead cheetas should be better protected in the wild.
No, Americans should not keep cheetah as pets. This would create a dangerous situation in neighborhoods where children and other pets live. People could get hurt because so stupid person could not figure out not to keep a cheetah in his backyard. These animals are not pets even if there is a limited number of them.
No, Americans should not keep cheetahs as pets to keep their population up. Cheetahs are wild animals that would not enjoy lives in captivity in human dwellings. Being wild, they would also be prone to unpredictable behavior and may cause injury of grave deaths to the humans attempting to house them.