With that in mind, I wonder how much of the initial problem of "significance" is really something that should be solved by no longer using the mathematical concept in scientific contexts.
Both the Nature commentary and The American Statistician issue were directed at scientists, not the general public. @Shewmaker is correct that science journalism is frequently misleading and sometimes simply incorrect, but that is a separate question. The sad fact is that outside of physics, many otherwise perfectly competent scientists have an imperfect understanding of statistics--what they mean and what they don't. I think it's healthy that a correction may be forthcoming.