It's not about the legitimacy of the tweet or even factualness.
When journalists want to create a fake narrative, what they do is say "people are saying this" and then show a few tweets to support them. Even though almost nobody really may be thinking in those term.
For example, there's a lot of social justice clamour over things like award shows lately. The thing is, people don't care that-that much. They may a little, they might have some nuanced ideas one way or another, but the 'journalist' reporting will say: "People are really upset about so-and-so's comments" - and then show a few tweets of either random people or celebs.
Individual tweets are not remotely representative of anything, and yet they are used as 'representative' data points all the time.
Not only do journos do this in purpose (i.e. willfully misrepresenting) but I think they are actually influenced by the Twitterverse. I really do think that many in the press think that Twitter is real or representative.