Ah, now I get it!
You were right, there is indeed still a main site (I guess I was thinking of the deprecated is_main_blog. Or something. Anyway I wasn’t getting it!)
My problem was that on my main site, I did not have SP activated. (The main site is sort of a gateway, which doesn’t do much itself, and doesn’t need a forum). So I tried activating SP on the main site, and that took care of everything.
Now, the SP updates appear on the /wp-admin/network/update-core.php page as they should, and updates there work fine. They even appear (and work) if I’m coming to the network/update-core.php from one of the other sites, not the main site (which wasn’t working before).
So I guess we can mark this one solved, and just remember that the answer is that you have to have SP activated on the main site, even if you’re not going to really use it there (just activate and then delete the forum page, for example).
Thanks so much for all the work tracking this down. What a relief!![]()