After some testing, including locating a high precision geodetic mark, it appears the reach reports coordinates relative to NAD83 when using the Indiana Department of Transportation CORS network. Good news, it appears to be dead-on for NAD83. NAD83 and WGS diverge about 1 meter here.
Height also seems NAD83 ellipsoid correct, which is about 33 meters below the geoid.
The height is easier to live with. It would be helpful if the json and shp files were labeled correctly (i.e. NAD83).
Please add my vote for supporting other GCSs, especially when communicating with a CORS.