They changed "On Device" to "Downloaded," and I don't know if that is really only for music downloaded directly from Google Play Music, or if it will play any locally stored tracks. Does anything come up when you flip that Downloaded switch?
EDIT: I just tested it myself by copying one of my ringtone mp3's from the Ringtone folder to the Music folder. Then I switched the Google Play Music setting to show Downloaded Only. Apart from songs that have been temporarily cached from my recent listening, the only other song that showed up was that ringtone mp3 (I otherwise don't transfer any music to local storage). So that option should work for you. If you don't see the music initially, try the Refresh Music option in the Google Play Music settings.