Sorry for the delayed reply.
The iTunes Store is primarily for people without an Apple Music subscription or that use Windows computers for purchases. I have a screen capture below from my iPhone showing the button you should see in the iTunes Store App that opens the music in the Music App where it can be played, added to your library, and downloaded for free. This is how both my Individual and Student subscriptions worked so we likely are on different iPhones and iOS versions causing some confusion, sorry about that.
I assume that button is missing for you and could mean your iTunes Store and Music App don't know about your subscription. I would first go to Settings > Apple ID > Media & Purchases > View Account. I have a feeling it will ask you to enter your password which might fix this. Once in there make sure to check your Subscriptions section that the Music Subscription is active.
If the Subscription shows as active and you are signed-in I would try to sign-out of the Media & Purchases settings then sign back in. If your previous purchases were made on other devices make sure that your have "Sync Library" turned on in The Settings App > Music. When you go to the Music App > Library, you should see them at that point.
