Your post sounds more like an Idea or a Feature Request. The best place for this is in our Ideas Lab. This way someone from WD will be sure to see it. However, I would search first and make sure that there isn’t another idea similar to yours that you could add your post to and vote on.
Can’t speak for music, but it does do this for video.
If you stop a video and exit (don’t just turn off the Live but exit back to the video menu) the next time you go to play that video it will give you a choice to resume play where you left off or not.
I don’t play any music but I’d guess this might also work – try exiting first before you turn off the power.