What you describe as "aha!" is Ni. Ne is the ability to see underlying patterns in everything on a subconscious level (which is probably why it is so difficult for Ns to effectively describe this process outside of its results), and analyze them, and I know from my own personal experience that it's a rare ability in the general population. It's not that the general population can't see them, it's just something that takes more effort on their part, something that doesn't come naturally and quickly to them.
I have a little Si, and if you are an ISTJ you have a little Ne. Everyone uses a variety of functions, but what makes you a particular type is your primary preferences and strongest ability.
I have no idea why some Ss mistype as Ns. I have a friend, for example, who totally mistyped herself as an ENFJ when she is obviously an ESFJ and has even told me herself that she prefers subjects which are factual, hands-on, and related to her daily life. I think it's because both are Fe dom, plus she believes she has latent psychic abilities, or "women's intuition" if you will, and that's how Ni - the auxillary function of ENFJ - is described to some degree.