I was thinking about tackling this problem just as an exercise, and I still might, but I think you’ve done some nice work here. My issues with this approach is having the music controller at the top. It makes sense if you want to make the point the iTunes is a music player at its heart, but that is less the case these days. I would use that space just for login/user profile, and search. But search would allow options for searching music library, Mac apps, iOS apps, etc., which would be accessed by a dropdown menu in the search bar.
Since QuickTime player and Apple TV put the controller/progress in the lower third, it makes sense to put the controller/progress at the bottom of the app. Or, just putting it below the main menu of the selected Media type would be better for me. Having the context of the UI change with the nav, the controller above it, and the content below it is jarring.
That said, I like your work here.