Okay, several things going on here...
First, you should NOT expect the transit lines produced in Add GTFS to a Network Dataset to represent that actual streets or other paths taken by the transit lines in the system. They are simply straight lines connecting stops and are meant to represent the logical connections between stops in the transit network, NOT for accurate visualization of the transit lines. Display Route Shapes, on the other hand, specifically uses the GTFS shapes.txt file and turns those into features for visualization on the map. Because GTFS has somewhat of a disconnection between shapes.txt and the files that represent the schedules, it's much much much easier for Add GTFS to a Network Dataset to just use straight lines.
Next, after you corrected those problems with your network dataset's settings, did you run Build Network and re-run Get Network EIDs? Without doing those things, the changes you made aren't fully baked into the network.
Finally, the non-traversable error to be looks like one of two things:
1) Your network has some kind of restriction attribute set up, and some roads are actually restricted (non-traversable). Perhaps the restriction is configured incorrectly.
2) You forgot to set up the layer's network location settings according to this section of the user's guide: public-transit-tools/UsersGuide.md at master · Esri/public-transit-tools · GitHub . As a result, sometimes the input points are being located on transit lines, and bad things are happening.