I don't know if there's really a disconnect there. Alderann was the "official unveiling" of the Death Star. It was the moment that the Empire showed every planet what would happen to them if they didn't fall in line.
But until the very moment that the Death Star was operational and the Empire knew for sure it would work, it was a liability. Imagine if the galaxy at large found out about it when it was 90% complete but not yet working. That would have been a HUGE impetuous for anyone who was unhappy with the Empire to suddenly go from being willing to keep their head down in the hopes of an eventual improvement in conditions to actively fighting against the Empire. There would be a lot of pressure to destroy the thing before it could destroy you.
And in context, even though the Death Star did fire on planets before Alderann, Alderann was the first "full power" shot. And when it comes to the other two firings:
Jedha was a backwater full of ruins and a religious cult. In addition, the Empire was already well established so they could have easily prevented any survivors from leaving (the Rebels had a strike craft that was prepared to flee).
Scariff was a major Imperial Installation, with major space defenses. Besides a few scattered Rebel ships, a majority of the Rebel fleet was decimated. Any other witnesses would have been Imperial crew, which Tarkin would have been MORE than capable of keeping under wraps.
As is, in fiction, you're looking at maybe a couple days between from the attack on Jedha and Scarif and the destruction of Alderann.