I have a feature service I have published from ArcGIS Server. They are 5 different trails feature classes from my SDE/SQL database. If I add them to ArcMap they show fine, just like they should.
They are the five feature services seen here:
http://gis.losalamosnm.us/arcgis/rest/services/trails
Now I have created an AGO map with them in it, and the features do not draw. This is the map:
Also if you try to open the table in AGO
I don't see the five feature services added to the map you linked:
The feature service in the map is a hosted one with no features in it. I was able to add the first few services from your Server to a new map, though, (the layer that's not visible is the hosted one):