This is the same issue I am having with my 2DS XL. I had the exact same problem with my DSi XL, and it was the upper LCD screen ribbon cable. It was damaged, and after replacing the top screen, it worked again.
I am waiting on parts to arrive, but I am attempting to replace my upper screen soon. I will report back once I replace it.
Because of the pop sound and shutting off happening in the exact same way on both systems, I am confident that it is the top screen to resolve your issue.