But all of the above new-found understandings and perspectives regarding the question aren't explicitly stated in the answer, but rather were a result of thoughts inspired by the answer making everything 'click'.
What would be an appropriate method, if at all, to share these new-found realizations?
Should I edit my original question? Comment on the helpful answer? Accept the answer but also add one of my own elaborating on all of the above?
I think adding a self-answer is the appropriate behavior.
Actually, I did so several times, e.g. here. I got an answer, the answer helped me to get a grasp on the problem, I then thought about the problem a bit more and in the end I had enough knowledge to type an answer that I would have given if the question would have been from someone else.
Sometimes several answers are given. Each one is good, but a combined answer would be awesome.
Sometimes an answer is given that indeed resolves your problem, but is not well written or took you a lot of processing to understand. If you can find better ways to explain it, then provide a self-answer. Some people will have your mind set and will find an answer written with such a mindset more helpul.
You might be "fair" enough to still accept the answer that was provided first instead of your "stolen" answer (I am not sure if this is the recommended behavior, your answer might be objectively better), but in the end, other people will come here and will upvote the answer which helps them the most.
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