If you really want to know, you can probably turn on flow instrumentation on better routers to see the traffic metrics. If your router does not have that capability, you could probably use a brute force approach and disconnect your Internet service at the router and see if the audio continues to play (not sure if the app will deal well with losing it's connection to the Internet).
I do agree that having playback information available in a user accessible diagnostic screen would be occasionally useful (like the "stats for nerds" in YouTube).