Interesting discussion.
Actually there are two distinctions here:
1. The medium used for the quiz? Do students read the questions on paper and put their responses on paper, or do they see the questions on-screen and enter their response with keyboard and mouse.
2. Are the responses marked by human or computer.
All 4 combinations are possible. (Remember these http://www.fotolia.com/id/39241911.)
So, you could say that the fundamental thing in Moodle is the on-screen bit, not the computer marked bit.
However, the computer marked thing is very important, because it opens up a new possibility: Giving the feedback instantly, when the student still has the thoughts about the question fresh in their mind. And that opens up new new possibilities like the Interactive behaviour, where if the student is wrong, we don't just tell them the right answer, we give them the opportunity to fix their own mistake.
So, computer marking is important, but it is not the only thing. You may want a computer marked exam where the feedback is still done in the old way: Everyone gets the feedback (if any) at the same time well after the quiz is taken. In this case the computer just saves you a lot of work shifting bits of paper around.
There is a spectrum between a quiz, which in the extreme case is a lot of short questions, and an assignment which in the extreme case is a single long essay. Yes, there are different tools in Moodle that let you choose how to do things at all points along that spectrum. One thing that the quiz gives you is the statistics report. That is some psychometrics to help establish the validity of your assessment. If that is useful to you, it is an argument in favour of the quiz.
Anyway, I think this shows there is sufficient case for the quiz to support human marking as well as computer marking, although of course it is not correct for teacher to use the quiz for every activity.