That’s somewhat overstated.
We know anesthesia "works," and we know some of its molecular targets, but we do not fully know the mechanism by which it produces unconsciousness, ie whether anesthesia eliminates experience, or mainly blocks memory, report, and integrated neural processing.
Anesthesia appears to be a fairly broad effect - anaesthetics work on plants, for example [1], even though they lack any neural tissue whatsoever. It would be extremely surprising if those effects were also targeted enough to halt only some types of brain activity.
[1]: e.g. https://doi.org/10.4161/psb.27886
The most important thing to know about anesthesia in the context of OP is that it often doesn't work. 'Anesthesia awareness' is real and probably more common than we think because anesthesia can easily produce awareness but block memory formation.
Keep that in mind when they make arguments about propofol... Which is one of the drugs mentioned in https://www.theatlantic.com/health/archive/2017/12/surgical-... and https://gwern.net/doc/psychology/neuroscience/pain/anesthesi...
https://web.archive.org/web/20120411063647/http://squid314.l...
"What did the doctor say? He told me that they couldn’t up the anesthetic because an overdose could cause respiratory arrest, and that it wouldn’t matter because the anaesthetic on any dose caused severe short term memory loss and whatever happened the patient would forget all about it. The second point, at least, was right on. One patient spent the entire procedure writhing in agony and screaming something incoherent to God. The doctor finished the procedure, took out the endoscope, and cut off the anesthetic, and the patient turned his head, looked the doctor right in the eye, smiled, and said, laughing “Wow, that wasn’t bad at all! Guess I slept right through it!”"