We're using current version of arcgis online with a very simple setup.
We have a custom group, that includes some of the Esri standard basemaps, plus 1 custom basemap (NY State Aerials).
We configured the organization Basemap Gallery to our new custom group.
We created an app using AGOL built-in WebApp Builder and configured it to synchronize with the organization basemap gallery.
When logged in to AGOL, we see the basemaps in the custom group, but viewed anonymously, (public) we only see the esri basemaps. All of them. Yes, the group is shared publicly.
David Coley, did you get a fix?