Ziggy Stardust said:
Perhaps, in the aftermath, the audience could see a lot of Imperial propaganda being put out about how crucial it was for Alderaan to be blown up, by describing terrible things perpetrated by its people against other planets, that, of course, never actually happened on the planet. Maybe, this propaganda about the system was already being made, before Alderaan even met its end?
I don't think Alderaan was on the Empire's hit list until Leia was exposed as a rebel spy. Tarkin seems to have made the decision to destroy it only when Leia resisted interrogation. Had Dantooine not been so remote, he might have backed off, but as it was, some well known planet was going to be blown up sooner or later to prove the Death Star worked.
Post destruction propaganda is more likely, or else the Empire was simply going to rely on the fear factor that a relatively peaceful planet was obliterated without mercy. With the senate dissolved, nobody was in a position to denounce anything.