How did you make it through this entire episode, and a mention of Jay Rosen, without referencing the "View From Nowhere"?! He has stated, and I agree, that the VFN is really the root problem, the unrealistic ideal that simultaneously dumbs down journalism and sets up unrealistic expectations for it in the public's mind.
The value of a piece of journalism, and journalists themselves, is ultimately tied to authority. Today's media are challenging the source of that authority, and Serial is but one example of that (see also: bloggers, Twitter, YouTube). For most of American history (or at least how it looks from the perspective of today), journalism held an unearned, sort of assumed authority, based on this nebulous concept of "objectivity" and journalists saying "you should believe me because I am a sterile conduit for information who calls him/herself journalist" and that because they could zoom out infinitely that they were giving you the "whole story".
Now people are starting to see that this unearned authority is hollow and doesn't always provide us with the most useful information. We are (I think) entering a time when authority is earned by, as you said, transparency, but also through experience, accuracy and verification, comprehensive (but not always equal) consideration, honest assessment of reality, and ultimately using the expertise gained by years or decades of exposure to a beat to deliver information that is useful and valuable. All of this rests on transparency, though.
And before anyone starts screaming "authority fallacy", I don't think this qualifies, because it is an earned authority, it is applicable expertise, and therefore has value.
"Objectivity" doesn't even have just one useful definition, at least in the public's mind. Here I'll rip off a few from Rosen: Should claims be based on verifiable facts? Yes. Should journos report what is and not what we want to be? Of course. Should it mean pulling back and showing where journos are coming from on an issue? Yes. Should journos be able to describe without offering opinion? Yep. But should should journalists only describe without offering opinion? That is where objectivity fails.
Journalists establish a much more real authority by being honest about their process, their knowledge, their experience, and their POV. This is essentially what you said about transparency, but without the "A-word". Koenig didn't do this perfectly (her white privilege is a failure of POV, her failure to contact Urick is failure of process) but she did it openly. Mostly.
Jay Rosen also makes a point that American journalism is in almost every way dumber than the journalists who make it. That is a sign of a broken system. Koenig's transparency and openness wrt her process and research (and you alluded to this) allows us complete access to what she knows, the journalism she produces is no dumber than she is. It may not be perfect, but it is a reflection of her knowledge, her process, and makes her authority (even if it is limited) very clear.