Another observation: if you take an alternative turn and get rerouted, the progress bar in the navigation page is reset and starts from zero. I can understand how that can come about but it is somehow misleading.

I think the easiest solution to that would be to assume that
progress = (distance traveled from start + distance to destination) / (distance traveled from start)

That way, the app would not need to remember the entire route and can easily throw it away on rerouting. It only needs to remember how much it traveled since the press of the "Begin" button.