Reporters are endlessly interviewing newsmakers and then using, at most, a handful of lines out of thousands of words. The paper, of course, may not have room for thousands of words of interview transcripts, but the Web certainly does.Now one comment on PolicyFrog's blog does point out that this would require a lot of effort transcribing the interviews. However, most reporters use audio recording devices to do their interviews; surely these could be uploaded to the website in raw form.
Nor does it make sense for the interviewee to give on-the-record interviews that are condensed into a handful of quotes: It's safer to have your full comments, and the questions that led to them, out in the open, rather than just the lines the author thought interesting enough to include in the article.
And for the institution itself, it's a no-brainer: You get a lot more inward links if you provide enough transcript that every niche media site can find something to point their readers toward. But no paper that I know of makes a habit of including transcripts of on-the-record interviews with major players.
Unfortunately, a lot of editors (or rather, nearly all editors) would balk at this, because it would provide a lot of ammunition for those who wish to point out journalistic bias (and let's face it, every news outlet has a bias of one sort or another). So don't expect it to happen any time soon.