Sleep does not simulate death. You are still aware of the passage of time, and can be woken by loud noises or bright lights.
Anesthesia is much closer to a classic death experience. You cannot be woken by external stimuli, even being cut. You also do not experience the perception of the passage of time. When you awake, you will feel as if the last time you were conscious was seconds ago no matter the time it's been.
Lovely that life-saving surgery is often facilitated by a substance that forces you to experience (or rather fully not experience) the thing you are trying to avoid.