I agree with your second point. If your intuition has proven to be false more than true (in a given context), then the intuition your brain produces would be evidence that the intuition is wrong (sorry if that was poorly worded).

As for the first point, I agree that it'd be nice to make an attempt to figure out what it is, but if the attempt fails, I don't think the observation that "Person X reports an intuitive belief that Y is true" should be ignored as evidence.