Followup:
I spoke to a Senior Advisor, who was very helpful on the phone. He guided me through the process of sending diagnostic information for Apple engineers to review, which I did. I received a call a couple days later, acknowledging that there was indeed a hardware problem, and that I should bring my phone in to an Apple Store for a fix. I made a genius bar appointment at the Apple store in Los Gatos. Upon arriving, the employee reviewed the notes and informed me that the identified "fix" was to purchase a replacement phone for $300. Also, they did not have any 128GB phones available, so I could have a 64GB from that store. At this point, I am more than a little frustrated because this is not a fix at all. Why did Apple send me back to a store for nothing? The employee felt bad, acknowledged that the situation was poor, and recommended that I call AppleCare back and ask for an exception. (His own manager refused an exception.) AppleCare refused as well.
I asked if Apple would recycle my current phone for a store credit to cover the cost of the replacement. They would recycle it for $220. I asked if they would at least meet the difference to cover the $300 replacement. They would not.
So, my options after buying a top-of-the-line iPhone in January 2015 are now to attempt to fix it myself, or buy a replacement phone. Extremely disappointing.
Most commonly repeated phrase through the ordeal from multiple lower-level support people (who were all very kind and personable.. but just unequipped to help): "I'm sorry, that is really frustrating. I feel bad, but there is nothing more I can do."
I spent five years working for Apple in the early 2000s. I've spent at least $20k on Apple systems in my life, and sold much, much more in Apple product to family and friends. I've always believed in the company's commitment to quality and customer experience, and I was always willing to pay a little more for that differentiated service and quality. I never, ever thought anything could push me to leave the Apple ecosystem. But with this experience, Apple managed to prove to me that they do not care about losing individual customers, and have lost my loyalty.