Also complementing Anonymous Mathematician's answer: one of the main uses of the "peer-review system" is "allocation of a scarce resource", namely, academic positions and grants. For that matter, it is not really accurate to talk about "peer review" in that regard, because typically it is not one's peers but one's "seniors" or "competitors" that reject one's paper, whether or not with good reason. But it is pleasanter to talk about "peer review" than "gatekeeping".
Another corruptive influence is that journals themselves vie for "reputation", which drives an odd jousting-for-position between authors and editors, wherein each tries to acquire as much status as possible, by connecting to higher-status "others"... Let's ask ourselves whether the same impulse wouldn't corrupt nearly any publication-for-status structure. I have no idea how to design-against, or design-around this.
If the status rewards are reduced beyond a certain point, we might indeed find ourselves with the "hermit gatekeepers" who simply have nothing better to do... as mentioned in the comment by John Moellner. Given human nature, it seems that the broad pattern demands incentives that are not entirely idealistic.
As it is, we can easily see that on Math Overflow, MathStackExchange, and many such vehicles, the most-popular questions are approximate repeats of questions that have been asked many times before. For a year or so, such a site will have a feel of freshness and novelty, but it is not clear to me what happens after some years when the limited repertoire of basic questions (motivated by undergrad or grad courses and such) is fairly completely documented, so that one no longer asks a question, but first carefully does the archival search.
I would advocate that academics who are already established, tenured, etc., think more in terms of the subject matter itself and putting helpful things on-line, rather than continuing to take up quite so much space in the game that beginners must play to get a position, get tenure, etc. True, one won't maximize one's pay raise each year without continuing to play the same game forever, but academic departments could really work harder to figure out how to appraise post-tenure people, which wouldn't upset the tenure criteria per-se at all. "Get out of the young peoples' way."
Edit: In response to "comment": I think the sense in which these points address the original question is that, if there were no presumption that the present "peer review" system should be supplanted, then why ask the question at all? Just use the "stack exchange" or similar software and everyone who wants to put papers or reviews of papers on it, and vote, or down-vote, can do so. There is no obstacle to this, so why ask the question... except to propose also that there be a transition. My remarks above, and some of the other comments, indicate why I think there would be problems. E.g., many of the people one would want to participate would not, for a variety of reasons, which would certainly have the effect of failing to confer one sort of status of a sort currently essential to professional survival at a junior level.
That is, putting the question in a context in which it makes sense to ask it (rather than a more formal sense somewhat disjoint from that context), the other parts of the context prove to be significant in answering the original. If one speculates that the issue is about replacing "peer reviewed traditional journal publication" (paper or electronic) with a different structure, then it is reasonable to see why there would be trouble.