I have the same worries! I prefer smaller local clinics but most of these don't have an anesthesiologist. I don't know what surgery you are doing, but the majority of cases of eye and nose surgery in korea are done under twilight anesthesia, not general anesthesia (unlike US and europe). You absolutely need an anesthesiologist for GA!!
For sedation, I'm unsure... I heard that even if they have an in house anesthesiologist they only do GA and the sedation is done by either a nurse or the surgeon themselves. So it might not even make a difference. On the other hand they would at least have an anesthesiologist available if there should be any complications. If heard stories of people waking up from sedation because it was too weak.
Many of the clinics who have results I like don't have an anesthesiologist and only do sedation... I feel like it should be safe given how common it is, but as someone with little medical knowledge it's hard to say.