Hi,
Personally, I'm rather irritated by the spread of Twitter as a replacement for basic communications.
I tend to agree with you, and it irritates me to no end when an organisation asks me to contact them on Twitter. The FSFE doesn't do this: our main contact points are email and phone.
But if we want to talk to someone, we sometimes need to use the channels of communication a person or organisation prefer. It happens fairly frequently I need to use Twitter for this, especially when it comes to reaching out to individuals: very few feel comfortable posting their email address publicly, but are easy to find on Twitter or Facebook.
At the same time, this is different from the "social media presence" which was discussed previously. I do think we occasionally need to communicate with people using social media platforms, but I'm not convinced we need to invest a lot in having a presence on social media, regardless of if that social media happens to be Twitter, Facebook, Ello or Diaspora.