I thought I'd give Apple support another call. Each time I call in I get a different response.
A few weeks ago, when I first called support for THIS iMac (returned the previous one for same issue), they gave me the vague answer that there would probably a fix in an upcoming version of Sierra. This past week, a different support rep told me that this was something they'd never heard of before, and forwarded my information to the Apple Engineers. Note** She suggested that going through support would be the ONLY way they would catch wind of the problem, so I hope everyone here has tried to go that far.
She then told me to book an appointment with the Geniuses--but after reading this forum, I see no point in playing with random hardware replacements. I already replaced this iMac once, and the problem only appears the moment I upgrade to Sierra.
I made the decision to extend Applecare in case this does prove to be hardware (probably their plan all along). I really hope they figure something out in the next two years. That should be ample time, no? Considering the high-end Windows rig I could have for what I paid on this iMac, I'm really starting to question my decision on choosing Apple.