Same for me as well.

I recently got an HDMI splitter and usb-c capture card to start recording TV. HDMI splitter downgrades the HDCP of the firetv stick so the capture card can record it. You might be able do the same with an apple TV as well. Have it play a playlist of music while recording then split the recording into the separate songs and get rid of the video. This isn't as nice as actually downloading the music but I'm not sure of an easier way using those services.