Or maybe, as I have already answered ircmaxell, Stack Overflow could raise the bar for asking questions ever so slightly from being the one thing you can do without rep?
If torrents of new users asking stupid questions are the biggest problem on the site this seems like a first obvious step.
I think that if you do what you propose, the site will just freeze in terms of user base.
It just would not be worth any new users time, especially given that when a user wants a question answered, it tends be slightly time sensitive.
One can easily overcome that with ten skillful edits, 2 upvotes (or 1 upvote and an accept) on an answer, or 4 upvotes on a question (20 rep is required for chat).
It's fairly easy for an individual to get into the chat, if they want.
That barrier does keep spammers from flooding the chat, though. And if you consider it an interview process, that's what the earning of reputation points accomplishes, and I'd argue it does so rather elegantly.