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One other significant issue is, how will the journals be indexed and accessed in the future? Scientific journals aren't like magazines, where you read them once and then throw them out. They are used for decades. And there has to be a way find the information, among the hundreds of thousands of articles in the scientific literature.
I'm a chemist, and I can't just google to find the research I need for my work. I need a specialized search engine that allows me to search for chemical structures. A few weeks ago, I used a journal article from 1917 - the only reason I was able to find that article was because the structures of all the compounds referenced in the article were indexed and available for searching. If an article is published on someone's web site, it's unlikely that it will be indexed and made available in that way, either today or 10, 20 or 50 years from now, for future scientists.
There are some major disadvantages to the current system, but what works for the latest "press-friendly" research might not work so well for other types of scientific journals. And while it's important to have the information today, it's also crucially important that the information is available in the future - and in a format that can be searched and accessed.