This scholarly link with a study on mice may be helpful to you. From the summary:
“In contrast with published actions of propofol, for which no REM sleep rebound occurred, REM sleep rebound occurs after exposure to volatile anesthetics in rodents, suggesting that volatile anesthetics do not fully substitute for natural sleep.”
This slightly-less scholarly (but much more lay-accessible) article by Psychology Today says that anesthesia is closer to coma than sleep, and states that comas don’t count as sleep:
"The brain works through information flow. Anesthetics produce amnesia, among many other things, because the brain cannot process the information received from the rest of the body—and communication within those parts of the brain that are partially active is multiply blocked.
“Information processing is what the body does. Information comes in, is processed, remembered, or forgotten. Much of that process of remembering and forgetting occurs during sleep.”
So, the consensus seems to be that anesthesia cannot substitute for sleep. This can be anecdotally confirmed by people who experience sleep-debt after surgery (although the physical trauma caused by surgery, as well as the aftereffects of the anesthetic, may confound that data). I’ve never been given an anesthetic, so I can’t personally attest to that. But it seems like anesthesia is comparable to a temporary brain-pause. Only the reptilian brain functions that are absolutely essential to survival go on during anesthesia. Whereas during normal sleep, your brain continues to actively coalesce the information you gathered while you were awake.