Some news about tweets is justified - for example the president of the US has at times announced policies on Twitter first. It's a modern equivalent of making an brief announcement to the press or sending out a very sparse press release.

And the tweets that you (and I) hate to see reported, those of random people and especially headlines like "Twitter goes mad about (whatever)" somehow making an entire article out of quoting 6 random Twitter users... it's a digital form of voxpops on TV or reader's letters in newspapers. All of them can be a nice addition at times, but in my opinion the vast majority shouldn't be part of a "news" medium.