I'm glad you asked this because I also had some smaller clinics tell me they would use IV Sedation and I was like huh? Because I had only ever see posts about general anesthesia.
I poked around the internet (most Real Self q&a area) and it seems that, at least in the US, surgeons prefer GA because it completely knocks out the patient so the patient can't move. With sedation, the patient is technically still awake, although out of it, but if they suddenly snap back to and panic, they can move around and that can be dangerous.
For GA, it seems a big reason that you need a dedicated anesthesiologist is because, as said before, GA completely knocks you out. This includes your ability to breathe, so with GA you have to be intubated/have a "protected airway." Under sedation, you're still able to breathe under your own power. Since the nurses and doctors will be focused on your surgery, you probably wouldn't want them also distracted by watching breathing monitors and such as well. In addition, an anesthesiologist is trained to identify if you have an allergy and can correct it on the fly.
You can ask the clinic if they have an in-house anesthesiologist, even if they are smaller or don't have one listed on their website. For example, I asked Dr Jin from Premium Nose if he has an in-house anesthesiologist and he told me they have a full time anesthesiologist on standby (meaning the person does anesthesia full time, but isn't employed only by their clinic). Even with IV Sedation, you can ask your doctor who will be administering it.
Another point, I think that most of the smaller clinics probably use IV Sedation because I imagine that anesthesiologists are expensive (both in house and full time on standby). I'm can't say for sure about Korea, but I have a couple friends that are anesthesiologists in America. One is a doctor and one is a nurse. They get paid A LOT more than their not anesthesia trained counterparts, BUT they also had to do a lot more schooling and training as well. The anesthesia programs are very competitive to get into as well and you have to be kinda a top doctor or nurse to get in (from what they've told me). Also, in case this adds an additional question: is a doctor or a nurse better? For plastic surgeries, either is fine. My doctor friend said the main difference is that the doctor version is trained to do more complex causes on the fly, for example accident victims that need immediate surgery in an ER situation. Also doctors can do research. Neither of these situations apply to surgeries we would be undergoing.