For me, the first functionality for the app would be to show the streets that I have not run yet. In map view, it could highlight those streets and also the parts of already runned streets that are missing. In list view, it could, using my position, give the distance to those streets.
A second functionality would be to propose an itinerary to go through non-run streets. The idea is that the itinerary would be computed or drawn from the website and sent to the smartphone and then we could tap on streets (either in map view or list view) to include or exclude them from the itinerary based on the actual position and the initial itinerary. This could be completed by a “performance factor” for each run (eg : if you change your itinerary to include streets A and B, you will have to run 1 km more and it will bring you 2% in city 1)
I run with a GPS watch and sometimes bring my phone (for pictures or for safety reasons). I use an android phone. For me the app should complement the website but not have the same functionalities.