Why can't I play my music without Apple Music trying to get me to purchase membership
I don't want Apple Music membership
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On Apple TV, the Music app is for use with an Apple Music subscription, as it is all cloud-based.
To listed to your music on Apple TV via Home Sharing, use the Computers app instead.
For iTunes on a computer, or Music app on an iOS device (iPhone/iPad), you can turn off the Apple Music features in Preferences/Settings.