Another indicator of public understanding of science focuses on understanding of how [scientists] generate and assess scientific evidence, rather than knowledge of particular facts. Past NSF surveys have used questions on three general topics—probability, experimental design, and the scientific method—to assess trends in Americans' understanding of the process of scientific inquiry.
Understanding of what it means to study something scientifically is considerably lower, at 18% in 2010. Correct responses on this question are lower, in part, because the task of expressing a concept in one's own words is more difficult than recognizing a correct response to a multiple-choice style close-ended survey question.
This is still much higher than I would expect based on occupation, since STEM [science, technology, engineering, and math] fields account for only 6% of the workforce. However, even though, as you say, "[m]ost people are not in a position to understand themselves and their own thinking", this is not insurmountable. Surveys similar to the NSF one I linked shows that over the past 25 years, the literacy rate has doubled (from 10% in 1988); clearly, the public can learn to understand rational, scientific methods.
Even if this conclusion is wrong, what do you think the proper method is to deal with the irrational nature of humans? Set up some sort of inner cabal of "great minds" to run the world (ignoring the fact they're just as human, therefore just as irrational, as anyone else)? Try to find some inhuman ("angelic") agent to run the world, and hope their goals remain humanly comprehensible? Or just give up and go back to the caves?