Jeremy K.,
I appreciate you getting back to me with this update.
To download music to your device, you should be able to hit the cloud icon next to the song, album, or playlist, which should then show a check mark next to the tracks to represent being added to your library: Add and download music from Apple Music or the iTunes Store - Apple Support. It sounds like that is not happening based on your description.
To confirm, when you navigate to Settings > Music > Automatic Downloads, do you have this turned on? When you add a playlist from Apple Music by hitting +Add, does it still appear in your Downloaded Music folder under Library? However, the individual songs are not part of your library unless you individually hit the + next to the songs?
If so, I would like to get you in touch with our technical support team to investigate this behavior in more detail. Please click on this link: Contact - Official Apple Support to connect with available support options in your country/region.
I appreciate the time you took to work with us in the Apple Support Communities!
Kindest regards.