Hi,
I also had my wisdom teeth removed in 1999, and had general anesthesia as well. It was a good experience, but I didn't have direct contact with an anesthesiologist as the oral surgeon administered the anesthesia.
However, in 2001 I had to have an abdominal surgery, and went into the hospital early one morning the week before Christmas. I was not very nervous, because I have an interest in surgery and anesthesiology as well. So I was taken to pre-op, and put on a gown and had my line put in with saline and everything. I met the anesthesiologist, and liked him immediately. He was friendly and personable and seemed sincerely interested in the people he was caring for. I only saw him for a few minutes before the surgery, and then a few hours later as I was being taken into the OR, he was walking beside the bed and talking to me, and I remember feeling very relaxed and safe. Or maybe that was the meds I'd been given! 😉
I was helped onto the table, and attached to a bp cuff and pulse oximeter and other things like that. The anesthesiologist was there, and still talking to me. I remember feeling that he was the most important one there, because I hadn't even seen the surgeon yet that morning, but the anesthesiologist had been with me for most of the time anything was happening. The last thing I remember before losing consciousness was him putting a mask over my nose and mouth and telling me it was the anesthesia and to breathe deeply.
Then I was waking up, when someone was calling my name. It was annoying, because I didn't feel like waking up at all. But it was the nice anesthesiologist, so I made an effort to stay awake. I think it took about an hour or more for me to be moderately awake and aware of my surroundings. I was told that things had gone well etc. and I needed to be waking up. And I didn't see much of the anesthesiologist after that, but the thing I remember most about that surgery was how nice he was, and how calm he made me feel. I can imagine that for people who are really apprehensive about their surgeries that someone like the anesthesiologist I had could be a great comfort to them, and make the process go a lot smoother. They are definitely a big part of the surgery team.
However, I must say that I do not like waking up from anesthesia because it felt to me like I'd just gone to sleep and needed to sleep a lot more, but was being asked to wake up. I didn't want to wake up, and it was a lot harder to try to stay awake than it was simply to give in and go back to sleep. The sore throat from being intubated is not too fun either. 😡