Your concepts are correct to a certain degree since intuitions can be wrong. The early fight by the establishment against quantum mechanics as a non-hidden variable theory was in large part based on intuitions. It just didn't seem to many that our intuitions could be that wrong as portrayed by quantum mechanics.
However, almost everyone wants the world explained to them in a manner such that their intuition will give them the correct answer. Now doesn't that strike you as a rather stupid and short sighted theme to use in trying to understand the world. It seems to me that any rational person would recognize that, in view of the facts listed above, they want something which most probably can not possibly be achieved.
As stupid as that might seem, intuition is largely responsible for selecting the mathematical axioms that appeal to humans as rational axioms. Even the logic of mathematics is largely based on intuitive notions. If, for example, multi-valued logic (e.g., true, false, not-true and not-false, less than true but greater than false, etc) wasn't so anti-intuitive, then I expect that multi-valued mathematics would also be popular. The point being that our classical minds are still mostly holding to traditional aspects of logic and mathematics, which is intuitive based. Intuition is inescapable since intuition is human thinking, and as long as we are humans that contemplate reality, we must do so on some intuitive level.