"Isn't the whole point of the exercise to acquire information about the world? That is the point of science, to observe, to gather information, test information and make predictions."
What leads you to think that mystics would have as their only goal acquisition of information about the world in the same way that science does ?
Is the aim of a weightlifter or gymnast or musician to acquire information about the world ? The only information of interest is mostly confined to what is needed to achieve their specific aim.
I doubt that a mystic would spend much time bothering to prove or demonstrate what they have experienced except perhaps for verification and exchange with others on the same path, since this is the realm of actual human experience rather than simply information about something.
Due diligence and standard of evidence would be related to their practice and the evaluation of students (if applicable) through their verification as useful of what was offered.
Also, a genuine mystic might write or talk a little of what they have experienced or understood, but the reason would likely be as a guide or confirmation for those sincerely wishing to take the path or perhaps as a fishing net. This attitude might well be an insurmountable obstacle for a curious scientist wanting to investigate or write a theoretical critique as an outsider.
Time and energy is extremely valuable for such people and they will use it only to further their own aims and help others on the same path. They do not engage in reality TV shows !
"So the story goes....however, if they are actually surpassing the natural limits of human capability ..."
Perhaps not expressed clearly, since I was not suggesting "surpassing the natural limits of human capability", but rather surpassing what is possible for the average human and in some cases would be regarded by them as impossible.
I'm not sure what you refer to with "could have been settled decades ago". What I refer to has already been settled a long, long, time ago. For example the ability to suppress or alter blood flow as required, or maintain normal body temperature in extreme cold or control heart rate, breathing and brainwave patterns etc.
Perhaps you are talking of "miracles" that appear to transcend natural laws ?
The point being that in the realm of what may be possible for a man, the only knowledge that counts is your own direct experience ... ie, real self knowledge .. possibly leading to knowledge of a higher order.
All else is hearsay, unverifiable or theoretical ... not entirely without use but not your own until verified.
The aim of any real teacher .. in any field, is to provide conditions and guidance for the student to arrive at his own verification.
