@PhilCFC: Sadly Samsung doesn't offer a music app that provides free downloads. If some tracks are missing form Amazon Music then they may have been removed from their service, therefore I recommend reaching out to Amazon support directly for clarification on this. As an alternative option, why not check out Spotify, which is free to use, however you will encounter frequent advertising as part of the terms and conditions of the service.
If you want to continue to listen to Spotify Music after stopping your subscription, you can try some 3rd tool like Spotify Music Converter to download spotify songs, album, artist, and playlist from Spotify without Premium and you still could listen offline.
You don't need to download the app. Nowadays there are many websites providing high quality music completely free like https://dzwonkinatelefon.pl/. you can find your favorite music and download it easily.
To avoid the issue you met while using Amazon Music online, you can try to save Amazon Music offline for using. If so, then you can use the Amazon Music Converter to help you. With it, you can convert and download Amazon Music to your computer as local files. Then you can move Amazon Music to any device you like for using offline.
As for music downloads from Amazon Music, the better tool to use is a professional Amazon Music Downloader, which doesn't require a premium Amazon Music account. It can directly download Amazon Music without apps and you don't need to purchase Amazon Music.
It can download all Amazon Music tracks, therefore you don't need to worry about losing audio files again. After that, you can transfer them to Samsung phones for offline playback without a problem.