I heard from Bluehost. The good news: they finally did the restore. The bad news: it doesn’t help.
But, this rep was slightly more on the ball than the others. She said that they had upgraded php, the scripting software the board uses, and that was causing things to break. It would have been nice to get some warning about this, or for other reps to be informed, but whatevs.
The obvious next step is to upgrade the phpbb software. I’ve downloaded the latest version and unzipped it, but I still have to mess with getting it onto the server. If you hear muffled cursing over the next few days, that’ll be me.
Edit: Looks like I got FTP to work. Kind of. Too tired to work on it now though.