Here's an update, and a possible explanation:

So I gave up on this computer last Thursday afternoon (I had other appointments and needed to get home). At that point, I left a chkdsk /f /r running because it was just taking so long. The user at that location reported that when the computer came back up, it wasn't much better, so they eventually swapped it with another less-used computer.

I followed up by going back to the location on Monday. When I got there, the disk usage was completely normal! The computer seemed quick enough and totally fine.

What I did notice was that the start menu had changed. It now showed the power, settings, and other icons as just icons (no text) shoved over to the left side. I believe this is the default for Windows 10 after the anniversary update.

The only thing I can think of is that these computers might be chugging away in the background on something as it tries to update the OS. I would have thought that the update would come through like any other one, where it downloads and installs, then does the bulk of the work when you restart the computer. But I'm not sure now. And if it is installing in the background, it's clearly not doing it correctly, and it's taking forever. I'm not sure what's going on and I wish there were some way to make MS aware of it. It's not like there's an error report box that comes up and says "I see your disk usage is unusually high - do you want to send an error report?" Heck, I've never really believed that those go to anyone anyway...

So that's where I am now. I haven't seen any real solutions to this problem, despite seeing people experience it all over as well as having experienced it myself over a dozen times.