What's wrong with a website? If something massive is happening, do you open twitter to look at selfbrags, burger pics, etc., or do you open your local/national newssite (cnn.com, fox.com whatever) to see what happened?
Want to see the weather? Will you open a weather site or wait for someone to put the report on twitter? Think you felt an earthquake... twitter? Or national eaqrthquake monitoring site? See a bunch of firefighters... twitter or emergency site?
"What's wrong with a website" is that it is not part of a relevance-curated, pushed feed that people are already using.
If you look at pre-idiot-king Twitter, when there's an earthquake it fills up with comments about it, and very rapidly. Geological advisories get retweeted, again very rapidly. It turns out that yeah, this kinda actually works!
I am not, I stress, saying that I like that this is the reality of things, but it speaks for itself.
