They did hand-wave the recording with the implication that Bail had to have been paranoid over his own death to have hidden it away on another planet; Even Leia expected it to have been destroyed on Alderaan.
And yes, the Senate is actually that corrupt. A significant fraction of the Centrist’s elite are Empire Apologists, and the book ends with them putting a man to death because they were given a nice, quick excuse to do so, rather than actually reading through what Leia had discovered, even if the man in question was the man who did everything he could to make sure Leia’s message would be heard.
It’s not a perfect book, but they do at least try to explain things.
EDIT: I think I remember why it was in this one Governer’s holding. I believe it was implied he was supposed to have been taken the place as Leia’s Guardian if her parents died.
Well, I’d give it a shot. Considering the way ROTS ended, it’s not too big a stretch to imagine most people are pro-Empire.