Stack Overflow is a Q/A system based on reputation. Do you think that SO will work as now if reputation was suppressed?
The main problem is trying to combat spam. Reputation is a way to do that, where by you have to invest time and energy into the system before you can perform certain tasks like up-vote.
It also is a way to create moderators and delegate out higher community functions like cleaning up, moving and closing questions. Essentially the SO community is paying users to work on the site by issuing reputation points. This keeps the user community clean and helps prevent spam from entering the system.
Reputation isn't about making you feel good, it is "a component of identity as defined by others"
While some people might not like admitting it, people like to be rewarded for what they do. When other people up vote your questions and answers it makes you feel like you accomplished something. Up votes are a kind of social payment for doing good that reinforce that you are doing something helpful.
Also, voting brings about competition to answer first and answer best as it allows you feel like you have done better than your peers when you get more votes for you answers.
I think this is a great question. How about no rep for a week, and see what happens? But the big problem is not the rep but the side effects - the key feature of SO to me is not rep/badges but self moderation, and it's hard to see that can work without rep. Still, I'm in favour of the experiment!