ISSUE SOLVed!
I ended up opening a ticket for this issue.
The workaround I was given was this:
I did the factory reset, then re-uploaded a track (without reinstalling the backup). The audio from the track survived the on/off cycle. But, after reinstalling the backup, the track audio died after the first on/off cycle.
I went with the second option they suggested - unmount the SD card (back it up on a computer if necessary), then reinstall it in the Stadium and do an SD format, then try again. I did that and this time it works - the audio keeps running after the on/off cycle (I did not put back the folders saved from the card, I just left it freshly formated by the unit). So far, so good!