there is still this question: what do you die from when you die from "lack of sleep"?
Here is a piece that explains some benefits of sleep. From learning and memory, cardiovascular health, resistance to disease, mood , safety, and metabolism and weight, sleep seems to affect us greatly.
But to answer your question, it seems several things can kill us from lack of sleep. But the underlying and principle cause would probably be exhaustion.
Just a guess.
Harvard Health