Thanks, everyone, for your help with my specific question and in general because I think this is an important topic, and I think the replies written in English are especially helpful for achieving clarity for native English speakers.
I am still confused somewhat, probably because there have been contradictory explanations in this thread, but it's becoming a little clearer, especially with these latest examples provided by Maître Capello. As to those examples:
I think I see why "emmener" is preferred in "J'ai emmené mon fils au zoo...," even though father and son stayed together, because going to the zoo is primarily for the sake of the child (i.e., the father is not already going to the zoo and bringing the child along with, he's taking the child to the zoo for the child's pleasure and accompanying the child).
The distinction between emmener and amener for taking/bringing one's child (emmener) vs. their parent (amener) to the doctor is a bit trickier to understand for me, as is the reason for preferring amener over emmener for bringing/taking a child to school in "J'ai amené/emmené mon fils à l'école."