how to get Apple Music songs and non-Apple Music songs on my iphone?

I have been collecting music for decades. Obviously, most of that was not downloaded through the Apple Music subscription. This includes a lot of stuff that is either not licensed to Apple, or is imported and not licensed for sale in the US, at all. It seems that if I want to select the music on my phone, I can either turn on Cloud music sync and download albums that are only available from Apple Music OR, I can turn off Cloud sync and download whatever music is on my computer BUT NOT anything that was downloaded from Apple Music.

Is this the case? How do I transfer Apple Music songs, and songs that I already own, to the same iPhone? I am tempted to ditch Apple Music and go back to buying albums, outright.