Random songs, playing via iTunes, that are not in my library at all or in any playlist

Whoah. Slow your horses there, with all this breach-this, malware-that and security-OHMIGOD nonsense.

Nothing in your post indicates that AT ALL. Are you seriously proposing that iOS has a security breach that some malicious hacker is exploiting so that he can play a random song on your phone? Not Gonna Happen.

In fact, I'd posit that you answered the question yourself.

If you subscribe to Apple Music then you have access to the entire iTunes Library. If you're playing shuffled 'Songs' then this may include tracks from iTunes Music that are not part of your personal library (emphasis is mine). Conspiracy solved?

I don't believe the security breach thing or blah, blah, blah, answers either! My problem & question is that after my selection plays, example: I select an entire album of an artist in a certain genre and when it reaches the end of that it starts 'supposedly' playing randomly and it plays songs from the genre and bands that I have in my library, but it is so narrow in it's choices. The same '5' bands out of a possible '25' or more, say, play and it keeps playing the same songs by the 'random' artists. Out of all Apple's music and my own library, can it seriously not play more bands and selections? Honestly?

This happens to me ALL THE TIME when I'm playing my own playlists. Not at the end of one, smack dab in the middle. Random songs from artists I've never heard of, never mind downloaded, show up in my queue. And yes, Autoplay is turned OFF. I hate it and am seriously considering ditching Apple Music for another platform that doesn't annoy me every.time.I.use.it.