The Copy Traversed Source Features tool will give a warning about empty output being generated whenever any of the three outputs (edges, junctions, or turns) is empty. This very frequently happens for turns. If your network dataset doesn't even include turns (the one from that tutorial doesn't), then you will always see this message. However, you should see something in the output edges and junctions feature classes. Do you see the layers in the map, and do they have something in them? They should. If not, please include more details, and we'll figure it out.
Regarding your other question about making the transit lines match the streets they run on: Unfortunately, that is not possible. LineVariantElements are constructed from GTFS schedule data, which is only loosely tied to the transit line geometries defined in the GTFS shapes.txt file. That file is often missing or of poor quality and is not suitable for creating a well-connected, routable network. The way our transit networks are built, the geometry of the transit lines is unimportant, as long as they connect properly at stop locations.
If you want to visualize your public transit lines, you can use the GTFS Shapes To Features tool, but this is unrelated to the network dataset and is purely for visualization purposes. If the network dataset LineVariantElement geometry matters to you a lot, you could attempt to manually edit the LineVariantElements to match the geometry of the GTFS shapes, but I think this would be very arduous and could generate a lot of errors if you accidentally made things disconnected.
Could you tell me why the geometry of the LineVariantElements matters to you? I hear a lot of requests for this, so I would like to understand your motivations so I can consider whether this is something we should support in the future.