iPhone Music Non-Streaming
Yes - there are three ways to accomplish this to have the Music app on an iPhone or iPad work without cellular connection and streaming data:


Disable the cellular entirely (remove SIM or disable the account with the carrier)
If you want data - just not for music - turn off cellular usage in settings for Music app
You can also just only play music that's cached.


All three of these need you to set up playlists and connect to WiFI and download those playlists / albums / songs so that the songs you want to be offline are cached. All of this data will come through WiFI and not cellular.

Now you can turn off everything (airplane mode) relating to wireless and WiFI and restart the phone. Open music and verify that the songs you downloaded will play in full. Now that you know how to cache - you can add playlists / songs / albums and cache them locally so you don't need to stream.

You don’t need another app d\to do this, but you might love the cesium player which makes it super easy to only see and play cached songs from your local library.