Here are my updates:
+ jumpmantnb, I already have autolock set to Never (I always lock manually), and I still do have the problem. I'm constantly locking my phone, though, so maybe it's the locking _process_ that's involved? I haven't seen that this is related to my phone being locked, so I haven't taken the drastic step of removing my passkey.
+ Scarlett, thanks for sharing your process. Because of what you've experienced, I'm not doing a factory reset on my phone to try to solve the problem - if it didn't work for you, I don't see a reason for me to go through the pain and frustration of doing it myself.
+ I had a Genius Bar appointment on Saturday. Chris in Seattle looked at my phone, and could find no evidence of misbehavior, after running all diagnostics and checking briefly in the logs. He insisted that, if there were posts in the communities, Apple engineers would be on the case. I asked if there were any mechanism where he could escalate his issue to Cupertino or something, and he said there wasn't, but not to worry. As far as I can tell, if the Genius Bar guys don't see anything wrong, there's no reason to believe that anyone anywhere is actually looking at this until/unless someone tells us otherwise. And there's no handoff between the people who are receiving reports of the problem and the people who in theory might fix it. Apple, if there's ANYONE paying ANY attention to this, it sure would be nice for you to assure us that there isn't a massive black hole in the communications chain. BTW, Chris did say that there was nothing wrong with my phone's hardware, and suggested a factory reset might fix the problem, and that, even if it didn't fix the problem for someone else, maybe it would fix it for me, it was worth a try.
+ I talked with someone who worked at Apple earlier. I think he had been an intern on the iPhone team. He said, based on what I reported, that it was a GPU overload issue, and a replacement phone might solve the problem.
+ I did test just leaving the phone alone when the touchscreen became locked. After a few minutes, it DOES fix itself, at least in my case. Which means that, even if my phone enters this locked state, there's almost no chance I can ever bring it to an Apple store to demonstrate the situation. If any of you ever have an opportunity to show it to an Apple employee in this bad state, please make sure they see it.
+ Scarlett, please let us know about your downgrade attempt to 8.0. At this point, my ideal solution would be to downgrade and not upgrade again until Apple acknowledges that this problem exist and confirms that it has been fixed. A 6+ phone should not experience these problems. I deeply regret having upgraded to iOS 9. And I'm puzzled, but glad, that my wife's phone is not having this problem.
