I wonder if you’re not confusing (deliberately, of course) “die” with “die out”. An individual organism does not die out, it dies, of disease, predation, accident or whatever. A group of organisms dies out. Such a group may include successive generations, so the term “die out” does not refer to individual death at all, it refers to a failure to replace those individual organisms after they have, individually, died. A simple failure to reproduce in sufficient numbers in sufficient time, caused by environmental stress, competition for resources, etc. is all that is needed for a group of animals to “die out.”
In the case of the rhinoceros, its lack of predators and availability of resources until recently meant that its population could be maintained by fairly low reproductivity, but recent human predation, very sudden in evolutionary terms, has not given it time to evolve adequate extra reproduction, and its population is declining to the point where no further reproduction will be possible, and it will become extinct.