There is no "Or, ".
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.