As a fan of science, I feel obliged to reply. I agree to an extent with your description of the methods and thus implicated shortcomings of science;
but I do not agree with the extent of those shortcomings, or at least the alleged source of the shortcomings.
In my opinion, science is the tool you say it is, and we have to accept it as one; not a tool to tell us the truth, but one to study it. Of course
this differs quite radically with your description of science as something that has the ultimate say of what is true or real. You are referring to the
way the collective uses science, so in a way you're completely right about it. But in my opinion, it's not science that's lacking here; it's us.
For example, when a newspaper publishes the results of a science experiment, they usually leave out some critical pieces of information (methods,
statistic significance of the results, acquisition of the population sample) - and 'rewrite' the conclusions to be a bit more spectacular. People
who quote the newspaper article often do the same, thereby reducing the actual information being spread once more. Eventually we end up with a lot of
blahblah and very little actual information.
When someone says 'science has proven X', I always feel the urge to say 'meaning; the hypotheses have not been falsified, therefore we hang on to
this theory'. Apart from the most exact of sciences, there is always some movement going on with the theories - that's how science works, it
expands. To quote wikipedia's hypothesis page:
Hence, failing to falsify a hypothesis does not prove that hypothesis: it remains provisional. However, a hypothesis that has been rigorously tested
and not falsified can form a reasonable basis for action, i.e., we can act as if it is true, until such time as it is falsified. Just because we've
never observed rain falling upward, doesn't mean that we never will—however improbable, our theory of gravity may be falsified some day.
We learn to understand one principle, then we learn to study the anomalies, derive principles from those anomalies, and there we go again. This is an
essential part of science, though often overlooked or misunderstood.
I'd also like to point out that science is not only experimental; empirical data is valid as well. For example, we can study a tsunami; we do not
have to reproduce it.
How does this connect to science studying only a selection of phenomena? Well, seeing as we get further and further, continuously extending our
theories, the box in which science is playing around will keep increasing in size; slowly, but very surely.
Of course your argument is that as a collective, we should have a tool that looks outside the box. My fear is that there is no way to 'construct' a
tool for this purpose. A tool consists of methods, and to be outside of the box, it has to be a bit more lenient than the scientific method. But what
does that mean? Is it a tool without theories? Without hypotheses? How could it build on itself -expand- if it has no such structure?
In the example of two persons experiencing the vision of aura's, a few decades ago we did not have the proper equipment to see what's going on in
one's brain. Now, we can definitely study those phenomena, and there probably have been quite some experiments (although I can't find them at the
moment). There is nothing preventing one from formulating a scientific hypothesis and/or theory explaining why some people can see aura's at some
times, why experienced colors differ, etc. Even if it appears that one of the subjects sees an aura at the time the other does not, and vice versa,
one could formulate an hypothesis to explain that. Even if that hypothesis gets falsified, one could formulate a bigger theory to explain how
sometimes the principle diverges from its normal implications.
Of course, the bigger the theory gets, the more hypotheses there are to be tested; and the less information we have (be it empirical or experimental),
the less conclusions we can draw - about the hypotheses, the theory, and most probably, the truth.
In the end, even though I believe the perceived shortcomings in science are really the shortcomings of people (both scientists and others) formulating
and interpreting results, hypotheses and theories. I guess we're in the same box; we feel science is being mis- and/or abused.