In our organization, the "Allow anonymous access to your organization" Security Policy is intentionally disabled. Although we serve specific content publicly for use in public-facing applications, we do not intend for our organization to be accessible for viewing and navigation by anonymous users.
Since 2019 we have used a custom basemap gallery group to serve public-facing WAB/EXB applications with our custom, publicly-accessible basemaps.
After moving to Experience Builder, we encountered [BUG-000167252: In ArcGIS Experience Builder, the Map and the Basemap Gallery widget do not display a publicly shared custom basemap if the 'Allow anonymous access to your organization' setting is turned off.]
and Esri have officially flagged this issue as "As Designed" and updated documentation with this note.
In order to continue to serve the public with our custom, publicly-accessible basemaps through Experience Builder, a significant amount of work is required to review and modify our organization site to enable anonymous access. We would like to avoid this by allowing the Experience Builder Basemap Gallery Widget to include the option to manually configure which public web map to use, much like how Web AppBuilder Basemap Gallery Widget used to allow.