File Geodatabases are not connected to the same as other OGR data sources. In the Add Vector Layer dialog, the user has to select Directory as the Source Type, then further specify the type (as Open FileGDB or Esri FileGDB), then use the file browser to select the top-level folder of the FileGDB.
When one moves files around, data may not be where QGIS expects to find it. On opening the *.qgs file, the Handle Bad Layers dialog appears. However, this opens a file browser that appears to be the same one used to add a file-type OGR data source, such as Esri Shapefile, GeoJSON, KML, GeoPackage, etc. As this dialog is not used to add a FileGDB layer in the first place, it does not appear to work in the Handle Bad Layers dialog. When you navigate to the top-level folder and attempt to select it, the file browser just drills down into the FileGDB hierarchy.
Is there a workaround for this behavior? How can I reconnect data sources (what the Handle Bad Layers dialog does) for Esri File Geodatabases when files have been moved and paths have changed?