1) Maybe you missed the "maybes" in my closing remarks.
2) Being exceptionally competent puts one in a social trap. It is generally not acceptable to state that you are that good. It is also not acceptable to lie, and downplaying your competence is a form of deception (aka "lying"). So if you are that good, there is no good answer when the topic comes up.
I will also take this opportunity to note that the surgeon did not come across as arrogant until I inadvertently questioned his professional competence by trying to verbally reassure myself. And, in fact, his reaction was extremely reassuring to me. I went ahead with the surgery with much relief and trust in his ability, something I don't think I would have felt if he had instead tried to nicely address my emotional concerns about the whole thing.
I hadn't missed your maybes. You're pondering the subtleties of this argument far more intelligently than my blunt comment does. But perhaps that's because you've had time to mature and reflect on your brush with arrogance, while I am still reeling from a recent encounter ;-)