Hi Brian,
I was able to tinker with this a bit today. I think I ran into the same problem you described and I was able to find a solution.
For me, I found that having an API key set prevented the app from challenging for the secured web map. With the API key defined (in App.OnStartup), the app would throw an exception that said I don't have permission to access the resource (without challenging me for a login). If I commented out the line that set the API key, I was challenged for username/password and the secure webmap loaded as expected. It seems that the app was trying to authenticate with the API key and didn't try to use OAuth at that point.
I think this works with the tutorial data because it's authenticating for a single (premium) layer rather than for a secure (private) web map item. In this case, it seems to effectively use the API key for the basemap layer and OAuth for the traffic layer. I think it's a little unclear how the API key and user authentication work together. I think this needs some better documentation.
Looking at the developer guide, I found a description of setting an API key's scope to allow access to private ArcGIS Online items. This is currently in beta and has some limitations. The API key tutorial also describes the steps to enable that access.
You might want to try following the instructions to scope your API key to access your web map and see if you can access it without using OAuth. Another option is to try just commenting out the line that sets the API key (in your project based on the tutorial) and see if OAuth challenges the user for access.
-Thad