A knife doesn't kill you, what kills you is the blood you lose after you get stabbed.

Lack of sleep doesn't kill you / does kill you in the same sense.

I'd probably kill myself after a couple of days without sleep. Would the lack of sleep be the cause of death or the cause of the cause of death?

Bullets don’t kill you, it’s the bleeding that gets you. Wait, no, it’s not the bleeding since you could just put an IV in, it’s the loss of blood pressure. No wait, it’s not the loss of blood pressure since we can reattach severed limbs that have been at 0/0 for hours. It’s the lack of oxygen to the brain and other vital organs. Bullets definitely don’t kill you /s