Why do we not reinvent the journal system?
Please meet the Open Access movement. 
In the last twenty years, many scholars and librarians did try to address the problem you pose. 
It is, in fact, a huge issue, and things like this don't change overnight. 

The OA movement focused on two main strategies:


Publishing in peer reviewed open journals (gold open access): the idea is to create a brand new journal (or to change the model of an old one) which will provide articles free for the readers, without the current subscription model in which libraries (meaning, taxpayers) pay. The crucial factor of Gold OA is the presence of peer review: organize real journals costs a lot of money, and at the moment the major business model is APG (author processing charges), meaning that the author (often, the faculty behind it) pays for being published and cover the journal costs.
We're still in transition, and there are a lot of drawbacks: there are predatory publishers who try to scam authors, and big publishers offer the "open access option" charging huge fees (this is also called double dipping, because a hybrid (both Open and Closed) journal will receive the money from subscriptions and the money from the authors. It is important to remark, also, that big publishers make a lot of money with subscriptions and they are actively challenging the open access model. 
PLoS, for example, is one of the new Gold OA publishers. 
Self-archiving in repositories (green open access): it is the model of arXiv, Repec, and thousands of other repositories. They can be "institutional" or "disciplinary", and they accept mostly pre-prints, but also post-print articles. 


Of course, there are some experiments in the field:


PeerJ has a very interesting business and pricing model.
Epijournals (from my understanding) are journals created selecting papers from arXiv and other repositories. 


References


Open Access, by Peter Suber, MIT Press, 2012.
Wikipedia "Open Access" article. Accessed 4 Jan 2015.


Full disclosure:
I've worked as a digital librarian in managing OA journals from University of Bologna. I'm biased towards OA and open knowledge in general. Please keep it mind that my answers reflect these bias.