The important thing to realize is that neither intuition rests on any particular piece of evidence. Instead, each one is a general outlook that has been formed over many years and countless pieces of evidence, most of which have already been forgotten.

This seems an overly rosy view of intuition that assumes it's largely forged of admissible evidence and competent updating. Is there much reason to believe that we tend to form worthwhile intuitions from sporadic bits of evidence?

I tend to be pretty wary of beliefs that have no easily accessed or communicated justification. What of epistemic hygiene? This looks like a time to confess your ignorance and move on.