Ever since Apple Music came around I've struggled with the Music app. I don't subscribe (never even activated the three month trial) because I prefer to buy the music for the handful of musicians I follow. Unfortunately it's very clear the music app is designed to push Apple Music.
Pre-Apple Music you could go into the Settings app and set it so only music you've downloaded would show. There is still a toggle that makes it sound like that's the case but in my tests it doesn't seem to do that.
This will remove all iTunes Match songs from this device, except for songs you've downloaded.
You see, my issue isn't really Apple Music. It's that all of my music shows regardless of my settings choices. Now, I can just go to the downloaded section every time (though I do find that annoying). But my 4 year old, who uses my old iPad Air struggles with that. If at home, connected to wifi, there is no way to truly lock it out of the music on our account. A couple times she's downloaded albums I hadn't put on there. Thankfully those times the albums have been okay but it's inevitable she would find one that wasn't as kid friendly.
That takes me to third party apps. I don't know when but at some point Apple provided an API so that developers could access the music you have downloaded. I looked around but couldn't find many reviews, and those that I did find didn't address my specific questions (or more accurately: my specific edge case). Finally I stumbled across one that would allow me to lock it to only music local to the device, Amazon Music. It offers an offline mode but still shows music you've downloaded in the stock Music app. You do have to log into your Amazon account. I don't know if being a Prime subscriber is a requirement (in my case, I am one).
Unfortunately you can't "delete" the music app or you cut off access to that downloaded content in the Amazon Music app (I tried it.) Still, what I can do is bury the music app in a folder on her iPad and show her how to use the Amazon Music app. She seemed receptive to it as she often was annoyed not being able to find her music more easily.
Doing this made me realized how much nicer I think the Amazon app is, both aesthetically and functionally. It has a dark interface, basic controls like shuffle are not hidden below the play back screen and it has swipe gestures for skipping tracks.
I buried Apple's app and have Amazon Music on my home screen now.