I found literally thousands of online complaints just like mine, with only one solution that seemed guaranteed to work: wipe the phone & restore it as a completely new one, then wipe & restore it from backup and pray the backup didn't re-install all the corrupted files that caused the problem in the first place.
This was a time-consuming process, so I took the laptop & phone to work with me so I could just let everything run for hours uninterrupted. During a quiet moment, I set everything up, took a deep breath, plugged in the iPhone...
...and before I could click on the "Restore" button, the damn thing started to back itself up & sync normally as if the previous several days of rage-inducing failures never happened.
Oh, sure, I had to do about a dozen sync sessions because iTunes would only download my apps 5-6 at a time and some of my data (amazingly little, to be honest) went to that big bit bucket in the sky, but by the time I was ready to head back home things were in pretty good shape... The solution having been as mysterious as the onset of the problem.
Now I just have to get used to the changes in iOS, figure out how to recover storage space lost to a couple of new Apple apps I neither want nor need but can't delete, and figure out why I lose about 10% of my battery charge per hour with the phone just in standby -- but at least it's recognizably a working iPhone again.
I also still have to figure out how to straighten out my laptop, but that'll be a whole new rant on another day. Right now it's 4:18am (got home at 3:00) and I'm supposed to be back at my desk before 10:00 for Weekend Manager duty (riiiight....), so goodnight!