Seriously though, you've got several situations and rights issues mixed together. In the case of the website, you can't rebroadcast performances without getting the music rights. Whatever rights were gotten for the original competition do not apply to any future rebroadcasts. The exception is a news broadcast where the music of the performance is part of the ambient sound. That's just news and you don't need music rights to broadcast a new story. However if an entire video of an entire performance is played as news, that's a gray area.
Though I don't see how it would be news. Maybe you'd see that on those morning "news" programs or the talk shows you mention, but that's not really a news story and replaying the entire performance is not really a news story. A real news story would almost always feature an excerpt only as part of telling a bigger story, not the complete performance.
"Cupcakes are bullshit. And everyone knows it. A cupcake is just a muffin with clown puke topping." -Charlie Brooker
I'm pretty sure this happened to Todd Eldredge once. He had to change out music becuase he couldn't get permission to use it.