Emphasis added on my error. Turns out Yahoo does let you download them, but it’s a bit of a process, and nowhere does it tell you that mbox files are what you’ll get. Yahoo has a “Get my data” tool. Go there, log in, submit a request, and wait until they notify you (about a week for me). They’ll send you an email with a URL, where you’ll go to download a .zip file that appears to contain most of the contents of every group of which you’re a member (the photos appear to be missing). Somewhat surprisingly, the .mbox files contain full email addresses even for groups of which you aren’t a moderator.
So, @gerhard, it appears I was premature in disregarding your suggestion–my apologies.
Edit: Yes, the .mbox process seems to work much better. Some messages are getting skipped (~100 for the apparent lack of a date, for example), but almost all the 38k messages made it, all the users made it (and a spot check indicates they’re all associated with the correct posts), all with the correct email addresses. It isn’t perfect at keeping topics together (the other script wasn’t either), but it’s doing pretty well. And, as a bonus, it makes for a simpler method to document than what I’d been trying to do. Only downside I see so far is the delay for Yahoo to make your stuff available to download.