At long last, after some extensive testing, I have answers to share as to what causes a song to stop playing. Raze was correct in stating that leaving the throne room before a song is finished causes it to never play again, but there is more to it than that. Here are my findings:
-Once a song is chosen, it will loop continuously, therefore there is no real "end" of a song to wait for.
-Therefore, you will always be in a situation in which you will be leaving the throne room while a song is playing (unless you do not start any songs to begin with), which will cause you to lose access to that song.
-Once you lose access to a song, it cannot be selected to cancel another song from playing and revert the throne room to a silent state. Therefore, there is a snowball effect where you slowly lose your songs one by one.
-Naturally, this is a nasty bug, as the loss of one song will likely cause a player to listen to another, only to lose access to that song as well, and so on until none remain.
-Even simply travelling to the other rings of the Battle Tower will cause the loss of songs.
However, it is also important to note that the bug is not only caused by leaving the throne room, but also by exiting the game. This is important to note, as I suspect the throne room is a popular spot to save the game before quitting out. Here are my findings regarding that particular scenario:
-Saving the game while a song is playing will cause that song to never play again upon reloading.
-This creates the same snowball effect described previously as a playthrough continues.
So, is there a way to avoid this bug? Well, yes, but it requires a bit of care:
-To avoid the bug, save the game every time you visit the Battle Tower BEFORE playing any songs whatsoever. Listen to the songs, enjoy them, then reload the save you made before you played them. This will ensure you do not lose any songs, but is impractical for normal gameplay.
-Of course, you could also simply not play the songs at all.
Personally, this saving and loading loop would cause me to lose my mind over the course of a playthrough, so I have the following advice to give to anyone that experiences this bug. Get the soundtrack or listen to the game's music on YouTube. Enjoy the tracks in-game while you can, but just know that you will eventually lose them all. I know that the Developer's Cut on PC comes with the soundtrack (at least on GOG), and the physical 360 copy of DKS comes with a soundtrack CD.
All testing was performed on the 360 version of DKS running on an Xbox One through backwards compatibility. I attempted to test this bug in the Developer's Cut, but my PC is too weak, and the game crashes whenever I attempt to load it. If anyone from the PC crowd would like to add to this discussion, or confirm that the same bug exists in DC, please do so.
I apologize for how long this information took to gather and report, but I hope it can prove useful to anyone that finds themselves scouring the internet for answers, like I did, only now they may actually find an answer, rather than nothing.