Specifically, I'm thinking about the story. It's not being covered very well, imho. Lots of missing information. Reporters could be applying pressure on more sources, not just the Rep and his aides. Anyone who pursued it as a tech story would be breaking new ground in journalism.
Assuming you're up to date on all that, now let's fast-foward five or ten years, when there surely will be a Twitter-like service with an editorial ego, like the NY Times. This came about in one of several ways. Possibly Twitter, Inc went public with a big market cap. With a highly-valued stock they could buy an editorial organization, shut down their other channels, and publish the stories exclusively on twitter.com, with links coming from anywhere you like (people could point to them from Facebook, Google, Firefox, Safari, whatever). Think about the power they'd have. Facebook or Google could do it too. Any of them have the economic power to become editorial organizations, quickly, through acquisitions.
Another option -- the Times could run their own Twitter-like service, at least on an experimental basis. Still publish links to their stories through Twitter. Doing this now would be smart, to be in the game when Twitter grows an editorial arm.
Another option -- new companies could rise out of nowhere, realizing that neither the Times or Twitter have the ability to integrate with the other. If history is a guide, this is the way it'll probably happen. (Groupon might be such a company, btw.)
Pause for a moment. You have, on your company's computer, all kinds of information about what was posted, when it was posted, from what IP addresses, and what client software was used. You have a copy of the tweet, if such a tweet ever existed. And you know for sure if it did or didn't exist.
You have all this information. You have an exclusive story. In every sense of the word. It wasn't given to you -- you own the story. And if you do the right thing here, you'll keep getting the exclusives.
And since the Library of Congress also has the information, if Twitter says no, they might have to say yes. After all we pay for all they do. In some sense, we actually own that data. No joke. If we don't why are we spending taxpayer money on it?