I did a quick bit of poking around and found that none of the permalinks on any of the sub-blogs were working. If I hard-coded a non-permalink URL (http://www.thesite.com/sub-blog1/?p=123, where "123" was an actual post ID) then I could get to the post just fine.
This seemed like a straightforward problem, but "wordpress multisite permalink" didn't give me an immediate hit. It took a little digging before I found this post. And even then, the solution was hinted at way at the bottom of the thread.
Did you activate the plugin for each site?Well, I didn't have a plugin that I thought was the culprit. But saving the permalinks on the site?
Have you resaved permalinks on the subsite?
I checked one of the sub-sites and permalinks were set correctly. BUT, I've seen stranger things in 23 years, so I hit "Save Changes" on the permalinks page anyway.
Well what do you know? It worked.
Not going to ask why or how. Just tucking this one away for the future and moving on with my life.