In our work we use specialized basemaps and we want them to be available to global users. Now such users have access to a list of all basemaps - not local ones of our organization as we want. We have already seen similar topics on the forum, but a custom widget from one of the topics does not give results. Maybe there is an updated version of it or another solution?
I did what you said and I published a blank map in the experience building which has all my basemaps that I configured for use in my basemap gallery. Here is the app, try and see if you can see it:
Yes... i see the same thing too. This doesn't make sense since all my items in the basemap gallery are shared, marked as authoritative, and even shared to open data. Question... do you have to use the experience builder? I know there is a way to customize the basemap in the web app builder to show your custom basemaps.
I think I have a solution for you! I created a web app in the classic web map and web app viewer where I customized my basemap settings. I copied the URL from the web app. I then created a blank experience builder and used the embed widget to paste my URL from the web app. It now shows my web app viewer with custom basemap selector in the experience builder!