The ideal schema approach is workable.
Alter Fields at first seemed like a possibility, but unfortunately is moot as there is a software limitation in that the user only has ArcGIS Desktop 10.0. If I recall correctly, Alter Fields was a new innovation ESRI rolled out in ArcGIS Desktop 10.2.
In regards to using the Delete Field geoprocessing tool on a shapefile, it does seem to have this functionality. At least, it works in ArcGIS 10.4.1 (see below):
Chris Donohue, GISP