Now look at Facebook. It's better designed from a visual and usability standpoint, and I think that gave them a little bit of an edge. But again, what really seems to have driven Facebook's growth is incredibly well designed social mechanics. Photo tagging is huge, for example. The News Feed is a huge engine for growth.
So, I think the answer is that it depends what you mean by "site design". A clean visual design, or even good usability may not be as important as a clean social design, or a clean API design.
As a designer you need to understand what are the key bottlenecks in meeting your goals, and design the heck out those.
Once you're Apple and you have a bajillion dollars, you can afford to polish the insides of your battery cases. But as a startup, you only get a few chances, and you have to spend your design effort where it really counts.
hilarious. and spot on. thanks.
While you don't need to have a very flashy and attractive website, you do need a clean and usable user interface, and stuff like clean colors and typography.
But if you were creating a craigslist for web designers, I would strongly recommend having a nice design - know your target market :)
For example, if you have a site for coders, having a technical but not completely clean site would be okay as long as it functions (service works as advertised). Of course these same users would probably appreciate good site design more later down the road. On the other hand, if you're making a site like amazon for the typical internet user, an organized site with clear call to actions may be best (good user experience). But later if you change stuff it might freak users out if you're not careful.
Not sure if my input really helps explain anything other than that each site is probably different (I know, I know, I'm a genius...) but I guess what I'm saying is think about how the users are using the site and why, or even better, go out and meet them to ask them.
So to answer your question I would release an early messier version to a closed group of beta users, ie. early adopters but before pushing onto the mass market I'd clean it up.
As long as your design isn't dead-awful, such that it makes users feel your product is untrustworthy or poor, release now and iterate. You might want to consider getting feedback, either from posting it here as an Ask HN post, or by asking your friends/family/strangers. See what they think, then make an informed decision.
The perfect is the enemy of good. So is the bad.