Unfortunately, there really isn't a good way to do this with current Esri software. You could try manually replacing the LineVariantElement geometry with shapes derived from shapes.txt, but you have to maintain good connectivity with the stops or the network won't work. This would not be an easy task.
Could you tell me what you are ultimately trying to do? Why do you need to visualize the actual on-street paths traveled by your Route and Closest Facility calculations? Some further information will help me recommend how to approach the problem and will give me some useful requirements if we decide to support this option at some point. Thanks!
Please keep in mind that the Network Analyst public transit-enabled networks are not optimized for passenger-facing routing apps. A lot of components necessary for a well-functioning app of this sort are not available, including the on-street visualization using proper shapes, as you have noticed. Rather, Network Analyst's transit capabilities are designed more for analysis and planning.