Great writeup - I've felt this way nearly my whole life. I have a different take on this - not sure if it was intended and no one else has brought it up: imposter syndrome is almost universal among intelligent and high-performing people who are not sociopaths or narcissists. Perhaps this internal drive helps these people continue improving and avoid becoming complacent.
That incompetent people who present well inevitably manage to find their way into organizations (and sometimes advance to a high level) is also common - I know doctors I wouldn't ...
The simple answer is observation.
As long as there's no difficulty swallowing or breathing then the reaction is local only. People with true anaphylaxis experience facial/tongue/mouth swelling and respiratory symptoms regardless of where the sting is.
Any respiratory or oral symptoms should prompt an immediate trip to the hospital.