Geodatabase feature classes rarely correspond to a single file on disk. If you open up a File Geodatabase using File Explorer, you'll see there's a large number of different files. So there isn't really a path to a feature class.
There is (sometimes) a path to the geodatabase itself.
If the feature class belongs to a file geodatabase (Geodatabase.GetGeodatabaseType() returns GeodatabaseType.LocalDatabase) , the Geodatabase.GetConnector() routine will return a FileGeodatabaseConnectionPath object. This object has a Path property.
There are other kinds of geodatabases, of course, including SDE geodatabases and feature service workspaces. SDE geodatabases may or may not be opened from a .SDE file, but feature service workspaces don't have a file at all, just a URL.
Hope this helps explain why it isn't straightforward!
--Rich