Why Wont Apple Admit That Apple Music Does NOT Work Without An Internet Connection?

First, connect to wifi. Then you have to go to the Apple Music app and click on the picture of the three little dots in bottom right hand corner. Once you do that, you will see a picture of a cloud with an arrow pointing down. Tap on it. This will download it onto your device from icloud.

See the Music topic of the manual

http://help.apple.com/ipod-touch/9/#/iph3e2e367e

also see:

Using the new Music app

How to download Apple Music for offline listening

Well, I don't have an iPod, but an iPhone 6. I added an Adele album to my library, downloaded the song to the iPhone. Turned on Airplane Mode (no wifi or cellular available) and then even went to far to power off and back on the iPhone. And the Adele album - which I don't own, I just added it from Apple Music - played perfectly.

Now not sure if the way Apple Music works is differently between iPhones and iPods.

I have no real idea how Apple Music works but if it were me designing it I would set up a way it would have to call home periodically to stop people from loading up a device with music and then never connecting it to the Internet, thereby preventing it from checking if the subscription was really still active. Daily would strike me as excessive.

Both my iPad (3rd gen; WiFi only) and iPhone (6s) are able to play downloaded tracks while off line without issue.I haven't owned an iPod in years, so cannot address that specifically, but have nothing in any documentation to indicate it would behave differently. Something you might try:

1. While online, open the settings app > Music and turn OFF both Show Apple Music and iCloud Music Library
2. Do a forced restart: Simultaneously hold down the Home and On buttons until the device shuts down. Ignore the off slider if it appears. Once shut down is complete, if it doesn't restart on it own, turn the device back on using the On button. In some cases it also helps to double click the Home button and close all apps BEFORE doing the restart )the Music app in particular)
3. Return to settings and turn the services back on.
4. After the library sync as finished, download a few tracks to test

You might also try logging out and logging back into your Apple ID under the iTunes and app store settings. And of course, you could make a Genius bar appointment and have it looked at.

But obviously this is not a general condition. some on your device specifically is not working as it should.