> might have shrunk as low as 30-40 cheetahs
Or, at one point a mutation, or a group of mutations, gave rise to a small number of super-cheetahs that interbred, and those cheetahs completely out-competed all other cheetahs.
Or, at one point a mutation, or a group of mutations, gave rise to a small number of super-cheetahs that interbred, and those cheetahs completely out-competed all other cheetahs.
The bottleneck consists in the fact that the present cheetahs are the descendants of a group of 30-40 cheetahs that have lived at a certain time.
There are multiple possible explanations for why the bottleneck happened.
There might have been a climate change or another abiotic cause that has killed the other cheetahs, there might have been a disease that has killed the other cheetahs, their prey might have disappeared, another predator, like lions or hyenas might have killed most of them, and so on.
Among the possible causes, both in this case and in the case of humans, there is also the possibility that there has been, as you say, some mutation that has given to a small group an unusually important advantage, allowing them to outcompete any other cheetahs, so in time only their descendants have survived while the descendants of any other cheetahs have disappeared.
In the case of cheetahs, such a cause for the apparent bottleneck seems much less likely than in the case of humans, because it is hard to imagine a decisive advantage of the mutated cheetahs. Something like one extra meter per second in maximum speed does not seem to give enough advantage over competing predators.
Ockham's razor says my scenario is more likely.
The one I'm most familiar with is of the integration of mitochondria into eukaryotic cells. My understanding is that all mitochondria seem to trace to a common ancestor, which would suggest that this mutation appeared once, and was so overwhelmingly energetically advantageous that all eukaryotic life descended from that single individual.
(There are non-mitochondrial life forms, including prokaryotic bacteria. But so far as I'm aware every complex multicellular life form, and quite a few unicellular ones, are all prokaryotes.)
That's not the point I'm emphasizing here. It's the overwhelming metabolic advantage afforded by mitochondrial ATP respiration.
Of that, it seems only one variant has survived. It either emerged only once, or out-competed all prior and subsequent emergences.