I think I figured it out!!!
It only shows "Play Later" if the FIRST SONG you start playing is while you are in "Album" or "Artist" view - NOT while in "Song" view. (Note this is on an iPhone 6S - but I'll be all iPhones with Force Touch work the same way)
Test this by:
1) Force quit the Music app
2) Start the Music app
3) Go to "Albums" or "Artists" (or even "Downloaded Music" -> "Albums" or "Artists" and tap a song to play it
4) Now, wherever you go, you can force press a song and see "Play Now" AND "Play Later" - even if you go to "Songs" or "Downloaded Music" -> "Songs"
And you can see where it DOESN'T have "Play Later":
1) Force quit the Music app
2) Start the Music app
3) Go to "Songs" or "Downloaded Music" -> Songs" and tap a song to play it
4) Now, wherever you go, you can force press a song and you WON'T see "Play Now" AND "Play Later" - even if you go to "Albums", "Artists", or "Downloaded Music" -> "Albums" or "Artists".
This seems like a bug. I'm going to see if I can report this somewhere.