Nothing gets logged when the basemap changes. Is there some other way to get this to work? Or is it not really called when the basemap layer changes?
So it appears that the BasemapGallery does not actually change the map's official "basemap". It removes the current basemap layer, but then just adds the new tiled layer to the map without updating the map's "basemap" (e.g., by using map.setBasemap()). In a way that makes sense in that you can add non-standard layers to the BasemapGallery, and those layers would not be one of the options available for the map.setBasemap method argument.
I can instead use the BasemapGallery's selection-change event to track the change to its basemap layer.
It would be helpful if the documentation in both the BasemapGallery and the map's relevant events/methods (basemap-change, getBasemap, setBasemap) noted that they are not talking about the same "basemap".