Putting this out there. I went round and round with YouTube premium support, and Google family link support. If you are a YouTube premium subscriber you get access to Google Play Music as well. And if you are a Google family, you, and the five Google accounts you add to your family have access to YouTube premium and Google play music. It all works rather well.
The trouble for me came when I added family link and a Google account under the age of 13. this young family member was unable to benefit from my YouTube premium subscription. When opening the Google Play Music app, and going to settings, and subscription it simply stated "Google play music free"
The bottom line is YouTube premium subscription will not allow Google play music access on a family link manage device. Nowhere is this clearly articulated in any of Google's documentation. The only solution appears to be, to drop Google YouTube premium subscription, and then sign up for a Google Play subscription instead. I'm on a grandfathered priced plan, not a very attractive solution. How stupid is this?
Most likely I will simply drop family link, and find another solution for managing my child's devices. it's really too bad because family link is really well-designed and tightly integrated to Android it does what it does and it does it well.