You are correct, and I think people forget what the Empire was all about at this point in the movie. Up until this point, they have been a growing force over the galaxy, crushing the last of any form of republic/democracy, and up to this point, a small "terrorist" group of rebels (they are "scum") has been threatening what is otherwise peace in the galaxy. In their belief, they have had the means to crush opposition's actions, but not their will, and thus a doctrine of fear of power instead of power itself was meant to take place. Thus, it was in the Empire's interest to take their time to destroy the Rebels, to further solidify fear for everyone, like some Friday the 13th/Halloween movie where the killer just keeps coming ever so slowly. Let's not also forget that, with the exception of Vader, just about everyone is pretty confident that nothing could stop the Death Star -- even the Rebels considered it a long shot and they only succeeded because they had a Jedi-in-training in their ranks.
So yes, the Death Star was for show, and it would have been a very effective show.
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