>The gist was that modern implementations suppress memory formation rather than induce unconsciousness. That you remain in some sense aware of what's happening but don't remember the experience.

That's called twilight anesthesia and it's used for some procedures, not others. Usually used for stuff like wisdom teeth extraction and colonoscopies. Anything "major" and you're getting general anesthesia. You can ask what type of anesthesia you will be receiving (twilight or general).

https://wikipedia.org/wiki/Twilight_anesthesia