The Geodatabase toolkit (GDBT) for ArcCatalog is a very useful add-in for ArcGIS Desktop. It provides a collection of tools that you can use to troubleshoot, monitor, and investigate enterprise geodatabases. It is a separate install for ArcGIS Desktop 10.0.

Geodatabase toolkit - Overview (arcgis.com)
The toolkit is only available in 10.0 and can only be made available for 10.1 and newer using some registry modifications.
However, as most workflows are now migrating to ArcGIS Pro, it would be best if the same functionality is available in ArcGIS Pro as well (Since ArcMap is going away soon).
Since it's a very useful feature for database admins, I hope this can be a supported feature in ArcGIS Pro, as opposed to a community add-in with no tech support.