Heck, flying a plane is definitely intuitive, but it takes a large number of hours to be considered good enough to do it on your own.
EDIT: The ultimate example is flying a helicopter. All of the parts of a helicopter are so dynamically linked, a simply described operation like "move to the right" is going to take a bunch of simultaneous adjustments on several controls at once. There is actually no way to fly a helicopter, except intuitively, and it's definitely not easy at first.
In your helicopter example, the interface is (your) intuitive because it can't be done by thinking, you have to just "intuit" what to do.
However, people would say the helicopter is highly counter-intuitive, since there are four main controls and four degrees of freedom, with obvious linkages between them to anyone who has a passing knowledge of 'copters or 'planes. Despite this, every control affects two or more degrees of freedom, which is highly counter-intuitive, since everyone is used to vehicles where the controls affect only one degree of freedom.
TL;DR: You are using intuition to mean "unconscious competence", but generally, intuitive is used as a synonym of "obvious."
Awareness of all of these different sorts of "intuitiveness" probably distinguishes the excellent UI designers from the average ones.
EDIT: Not only are the unconscious competencies very powerful, they also tend to be highly optimized.