I don’t see any updates regardless of version. I had one staging site that was still running Oxygen 2 something, and all other Oxygen sites had various versions of Oxygen 3. I had to find their status by using the Manage Plugins and searching for Oxygen because none of the sites indicated they had updates available despite having the Custom Dashboard PHP modified identically to that in the forum post I linked in my OP.
In order to update them, I wound up logging in to each site and doing it manually. In retrospect, I probably should have just done what I wound up doing on the staging site with Oxygen 2; pushed it from MainWP favorites.
As a test just now, I rolled back one staging site to 3.8 from 3.8.1 since I still had 3.8 loaded in the MainWP Favorites. Even after a manual sync with that site, MainWP gave me no indication that an update is available. It didn’t show the update when I logged into that site directly, though, so I went ahead and replaced it with Oxygen 3.5, which did lead to immediate detection of updates directly in the site.
I went ahead and ran a fresh sync on that site from MainWP, and for the first time saw an available update status for Oxygen (and Oxy Extras, which has also been elusive, but is off topic).
Maybe they will start showing up going forward. Who knows. I just found it odd that none of the updates showed in MainWP despite showing in the site dashboards, and thought I should ask here if anyone else has encountered similar. Thanks for the reply.