You're quite right; an "offline" mode would help make Apple Music just a little bit better! However -- as you found out yourself -- nothing like that is baked into iOS.
However, there is a workaround: create a smart playlist in iTunes instead. This method assumes that you have a computer with iTunes installed, that both your computer and iDevice are linked to Apple Music, and that you're connected to the internet on both devices.
To get started, open iTunes on your PC or Mac, click
click the small plus sign in the lower left-hand side of the screen,
New smart playlist....
In the new window that pops up, click the drop-down menu that currently shows
and simply use the default settings as shown.
From there, your iDevice should automatically sync with your new playlist. What's great is that the playlist only shows which songs you currently have downloaded to a particular device.
I hope this helps, and feel free to ask away if you don't understand something or run into any issues!