Currently the tool can add measurements to the attribute table, for ENVELOPE, these are
Two improvements:
I suspect that it isn't done by default is because this already exists
Add Geometry Attributes—Data Management toolbox | ArcGIS Desktop
and people may not want the extra baggage of information if they are only interested in the geometry.
So it would become a two-stage process for you
However
but I can see where having a checkbox to specify the outputs would be useful since width and height must be calculated from the Extent and the centroid is floating around there as well.
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