[Natsu]

It might be possible for him to hear about the broadcast by other means had he not been in the first region where it was broadcast.

But it never really turned out to matter, as you say. L was able to notice that weird case where the evil guy who had taken school children hostage died of a mysteriously timed heart attack and piece a lot of things together.

Actually, though, that broadcast was probably one of the most important puzzle pieces. He was able to prove that Light could kill remotely with nothing but a face & name, having cut a special deal with a criminal and kept the guy under special observation the whole time. Light was very careless there.

[Shaanie]

It would be extremely hard (probably impossible) to hide the fact that you need a criminals name and face to kill them. At most he could've gained a bit of time by being more careful, and at that point he didn't believe the death note would work anyway.

[Natsu]

Maybe if you assume that psychic killings are possible from the outset. While people might be able to figure out that something is going on, proving that someone is doing the killings remotely via a notebook is quite a leap and only the crazy arrangement L tried made proving that possible.