But it was already huge success when it had no business model. Moreover, what is fundamentally valuable about Twitter to its users--sharing and discovering little bits of textual expression over a publicly visible social network--is not very expensive.
From the perspective of the users who find it valuable, why does it need a for-profit model at all? Why can't we just subsidize it as a non-profit via grants and donations, a la Wikipedia? I'm pretty sure you could do the important thing that Twitter does--ignoring all the extras devoted to figuring out how to extract more money from the data--at a small fraction of its $2 billion in revenue.
I'm not being naive here--it's quite obvious why things are the way they are. But there many examples out there of making a big impact while making a decent living (just without anyone trying to become a billionaire). Social networking is ripe for more of this approach. The attempts so far have failed not because of their business model, but because of the usual reason: poor execution.