How would you go about showing that emotions, feelings and intuition produce good results without proving that very idea rationally?

While intuition may be useful to get good results (perhaps it is the subconscious quickly applying some pattern that has worked in the past) there's no way to know a particular use of intuition worked unless you go back later and rationally analyze the results.

And more importantly, there's no reason to believe intuition works at all unless we either 1) understand it on something like a neurological psychological level (I would not hold my breath waiting for this) or 2) do some statistical modeling of it's reliably.