The http://route.arcgis.com/arcgis/rest/services/World/Route/NAServer url is fine, but you will not be able to see the rest service directory for this page by just typing it into a browser as the service directory has been disabled. It will still work in your widget though.
FYI - The documentation for the ArcGIS Online Network Analysis Services is available at http://route.arcgis.com/, while the documentation for the ArcGIS Online Geocoding Service is at http://geocode.arcgis.com/.
I was just looking to get a route service that might have had more up-to-date data as I know of places where the road alignment has changed in the past couple of years and these are not included in the default route service data.
Could it be that this is a secured service, as it requires AGOL for Organization credentials, and both the OOTB and your modified Route Widgets do not support the use of a secured service as the widgets are currently coded?
That seems to be the case using the old URL did not require authentication and the new one does. As I do not have an AGOL account I am now unable to test/develop using the new url, so I guess this widget will become obsolete.