Theyde loose the ability to naturally hunt and over-trust humans.
Cheetahs aren't like pidgeons they're predators and the best place for them is in the wild.
The problem is this then that they go after life stock and be killed for it. In Namibia there are NGOs that for free, will take them off you.
Of course the problem remains you've lost your livestock and people want revenge. It's the same story for African wild dogs (perhaps the most successful predators).
Given that the alternative is their eternal absence, and the total loss of a species, options are quickly disappearing.
The best place is in the wild, but the worst place is as stuffed relics in museums. When their numbers dwindle low enough, I'd expect the species would be lost to unsustainable inbreeding producing compounded birth defects.
Incidental taming and domestication as a side effect of preserving the species might be better than no species at all. At least habituation to human contact is a reversible process. Rendering a species feral becomes an option, once their numbers are strengthened.
Captive breeding has been very successful:
"The centre first achieved international recognition for bringing the cheetah back from the brink of extinction ..."
Judging from the article it might be pretty expensive, one of the things they talked about was the space requirements which makes sense, cheetahs need to run. I think that's part of why this is such a tricky problem to solve.
Cheetahs aren't like pidgeons they're predators and the best place for them is in the wild.
The problem is this then that they go after life stock and be killed for it. In Namibia there are NGOs that for free, will take them off you.
Of course the problem remains you've lost your livestock and people want revenge. It's the same story for African wild dogs (perhaps the most successful predators).