You can actually disable anything you don't want (including apple music) from the preferences menu.
I open the iOS app Music. I see Playlists, Artists, Albums, Songs and Downloaded Music. The first four show iTunes purchases. Only Downloaded Music shows what's only on my phone. Tapping that shows another set of menus for Playlists, Artists, Albums and Songs and a "warning" that it's "Showing only music on this iPhone".
Having a second/replicated menu set is inferior to the former ability to simply hide/toggle the iTunes content that isn't on the phone. And Apple feels the need to use up some of the extremely limited UI on my 5s to warn me that I'm not seeing my iTunes purchases! The whole thing is convoluted, messy and a blatant attempt to push the iTunes Store.
The app doesn't offer a way to make the default Playlists, Artists, Albums and Songs be that downloaded version only. From the home Library it always requires an extra tap. Not the biggest deal in the world but stupid nonetheless and it highlights their motivations.
Sure it can be useful to be able to view and download my iTunes purchases directly from the Music player, but I don't want this function to be the central or primary view in the app which it now is. I rarely buy music from iTunes because the value isn't there when I can usually get lossless versions for the same price or less from alternative sources so my purchases are mainly older stuff and free singles I don't care about any more (or even to begin with in the case of free singles).
I want these iTunes purchases to be relegated in the app not elevated as they have been!
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Yeah I know the reasoning, it just sucks. Any tips for a music app alternative for your own music?
There are so many out there. Look for what's popular and try out some of the free ones. Also look for ones that have a good EQ which the iOS Music app lacks. There are also music players that support high resolution audio and other formats (like FLAC) which the iOS Music player doesn't. Some can sync or integrate with other services, even iCloud.
So basically the iOS Music player is crap and limited. I'm hesitant to recommend anything in particular as it's best to do your own research. I bought a player that supports high resolution audio as I have a high-res DAC and a few high-res albums. It was expensive and overpriced (the app that is) but I got it on sale. It's pretty good but crashes on occasion. I'll find some other alternatives first then post again.