Apple - How to disable Apple Radio in iTunes and iOS?

It seems that the answer is:

There's no way to disable Apple Radio from taking up space in iTunes and Music on iOS.

If this also bothers you, you can submit a report here: http://www.apple.com/feedback/

Or, specifically, here: http://www.apple.com/feedback/apple-music.html

I have found a workaround in that using another app to play the music you have from Apple Music / Music.app bypasses all streaming.

• Cesium Music Player by Mike Clay

The interface is refined / simpler visually than Apple's new interface and it lets you control a bit more of the UI than Apple's app does. I find myself using it to only play songs that are local to iOS on long roadtrips to avoid a large download bill and stuttering when I am out of data coverage.