If you can cannot find any facilities then try using the route solver. Can you find any paths between your one incident and any other facility point? Most likely there is some connectivity issue in the data or a restriction that is causing the path not to be found. If the route solver does not work, I suggest that you use the service area solver, load your incident as the facility and on the properties of the layer, polygon tab, turn off the polygons and on the line tab, turn on the line generation and solve for say 30 miles (or something appropriate) and see if the returned set of lines show any problem with connectivity.
To answer your other question on what is the most appropriate solver for your scenario, it is the closest facility. You can use it to find the roads that will be on the shortest path from all your incidents to all your facilities.