I think Vocal does something like that (pushes quality, or at least most-read, stories to the top). New Voices section is a great idea.

I agree that people do read short fiction - it isn't that I think literally no one reads it. But it's a hard sell, and I'm not sure most writers will do more on that site than pay for membership, post their stories, link to their stories, and wait for people to read them. Unless the story is being shared by someone else (word of mouth) or fits into a popular genre readers can't get enough of (fantasy/paranormal/sci-fi right now, I think?), I think it'll be tough to attract outside readers. How many people buy literary journals if they aren't published in them - or trying to be?