However, by the end of the show [I don't count this as a spoiler as everyone should know Leia's role in the original Star Wars movie by now], Leia is returned to her loving adoptive parents and will go on to be a Senator in her own right. It's already been established that her father was in contact with Obi-Wan. It's clear by the end that she was transported home by members of The Path. How does the Empire just forget or ignore these facts?
It doesn't.
I now believe that Alderaan may have been Tarkin's first choice to test the Death Star because he already knew that it was home to significant leaders of the Rebellion. I think that the Emperor waited until he had the power of the Death Star in his hands because knowing that the Organas were not trustworthy gave him an avenue of investigation until he was ready to strike decisively (much like how he waited to give Order 66). This explains why Tarkin was so entirely unconcerned about the destruction of such a well known planet--he knew that it was already on the Emperor's list of targets.
Now the plot hole that bothers me is why the Organas wouldn't have figured this out too.