Actually the leading cause of surgical and post-surgical mortality is from deep-vein thrombosis or embolus resulting in some form of fatal pulmonary or cerebral blockage.
That said, choosing local anesthesia combined with general sedation has far fewer risks than general anesthesia. Unfortunately, because of the speed and ease of modern general anesthesia, proper local anesthesia techniques are diminishing in modern medicine.
A few years back I knew a guy who could even perform many forms of thoracic surgery using local anesthesia. Amusing side note, said doctor was once married to Ingrid Bergman before she became a Hollywood actress. Anyhow, I don't know of anyone who would even attempt this now, as it's far too skill-specific and not time or cost efficient, even though it is often safer.
My own experience with wisdom tooth removal was becoming conscious during surgery, despite a fentanyl/versed combo sedation with nitrous chaser. I wasn't hurting, there was still local in the mouth, and I wasn't bothered because I was in that opiate "Heeeey how's it going?" sort of mood..but it startled the doc.