Each contributor gets recognition in each individual question, and the quiz has a team designation on it to show that it was created by the team.
The few bonus points given is simply a quiz-creator bonus. If the member creates the quiz, yes, all players may contribute, that is certain. However, as agony stated, they have the corrections to attend to, as well as any player messages that may arise. The author of the quiz itself gets the few bonus points as a small reward, so to speak.
It's been brought up before, not feeling it fair for the author to get the 2,000 points and not the entire team, but as both a quiz-author and a team-quiz contributor, I can tell you it's definitely harder to be the author, and a reward for heading a team quiz, which can become extremely difficult and frustrating at times, is appreciated.
Subliminal message being sent. Did it work?