Every major website should have their own URL shorter. People would be more willing and trusting to click a random link found on Twitter if it's from Gizmodo's short URL rather than bit.ly.
"Then again, people shouldn't click random links."
What does this even mean? The web's primary use case is clicking random links as defined in some context. Shortened links rarely show up in a tweet without any defining context.
No need to build a new one every time; services like http://totally.awe.sm let you set up a URL shortener on any domain. (Disclaimer: the founder is a friend of mine)
And before you say "it's trivial, I could build that in a weekend" I suggest you try it; there is a lot of hidden complexity in something so simple, especially at any kind of scale.
