Some parts of this TV program should be re-edited before broadcasting it to THE public.

Airing and broadcasting are always done to the public or an audience (I guess it could be a private group, but a TV program is generally broadcast on TV for all to see). With that in mind, each of your sentences would be just fine if you eliminate "to the public", "it to public" or "it to the audience". Including these phrases is not wrong, but it is unnecessary. If you edit the four sentences in this way, I have a slight preference for "aired" or "airing" versus "broadcast" or "broadcasting" since I believe they are more everyday terms, but any of the four edited sentences would be just fine.

I think the first is the best, because it is grammatically correct and it is the clearest. "being aired" clearly refers to "some parts of the TV program", while in the other sentences, I don't think "being broadcast" is correct (tense wise), and "broadcasting" and "airing" have subjects different than "some parts of the TV program", which is not clear in sentences 3 and 4.

I think the first three sound better than the fourth, but really between those three there is no real difference as to which sounds more natural. They all sound just fine, equally formal. I suppose maybe #3 would be the most natural if I had to select one, but not by much mind you.