What do you mean, "you people?"

"Before publishers accept an article one of the peer reviewers..."

What you are suggesting may make it easier for non-specialists to read a journal, but would have no practical value/be impossible. The purpose of these types of journals/studies is to communicate advances in the science/field to your peers not the general public. Having state-of-the-art work reviewed by non-experts wouldn't be particularly useful. Plus it would be tough to find anyone to actually do these reviews if you need to hit a stack of textbooks before you can make heads or tails of the article you have volunteered to review.

The papers aren't written to make you look smarter, the conciseness is a result of the fact that an article published in Nature has a 1400 word limit. So if you want to get published in the top-tier broad-scope journal, you have to explain why your study is of interest to a wide variety of scientists worldwide in 3 pages or less...

The newspaper problem stems from distilling a highly concise piece of work, where every single word has been chosen to make the ideas as brief as possible, and then expecting a journalist to accurately shrink it down further into one or two paragraphs. There is no real fix to the problem, the average journalists aren't interested in keeping up to speed on the cuttting edge of all possible scientific fields and the average scientist aren't that interested in writing for the public (although some are and they will submit secondary articles that focus on the principles of the research (minus all the math) to magazines like Scientific American, Astronomy, etc.). If you want to get scientific info as a layperson, newspaper reports are guaranteed to disappoint. Their only real use is to prime you on what to look for elsewhere in greater depth.

It's almost certain that the journalists never talked to the scientists at all... Nature comes out weekly and there is an embargoed press release that is sent out to media outlets with a short synopsis/blurb of this weeks articles. Science journalists look it over and see whether there is anything particularly cool for the science section this week (i.e. nothing too abstract like particle physics) and then write up something quick for that weeks science section often just based on the press release (they may or may not read the actual article, which are often aimed at specialists and can be a difficult read at times). Longer form articles in the week-end paper usually include actually contacting the guys who did the study, but if there is no direct quote from the actual scientist who wrote the paper in the newspaper story then chances are high there was no scientist-journalist contact at all, and chances are almost as high that the journalist did not read the actual study, just the press release from Nature (after all the study was just published today).