You appear to be confusing "attributes" and "values". By selecting the [Data Type] as "date" you are informing ArcGIS that this field or column will contain dates. You have not told ArcGIS what those dates are. If you were to add another field or column and set the [Data Type] as "text" would you expect Arc to populate it with the word "text"?
The [Data Type] is important because in GDBs (and many other database types) you cannot mix data types within the same field or column. With your "Date" data type you can enter dates. You could not enter the text "Date not recorded".
As suggested in the comment, Editor Tracking might suit your needs. In order to enable it, open Arc Catalog and navigate to your feature class.
Right click > manage > enable editor tracking