Ok, getting back to the original point of the thread,
There's one very good argument I can think of for using intuition as well as evidence and reason in deciding what is likely to be true. Ever try to solve a math problem for a real life application (say deciding how much money to budget for a task), then get an answer and say "wait a minute, that looks like it must be off by a few orders of magnitude", and then go back and find out that you left out a decimal point in the middle of your calculations? Reasoning, at least when applied by real people in real life, is fallible, and sometimes those errors can be caught by intuition. And you already made a case about evidence occasionally being fallible with the example of UFO sightings.
So when it comes to the real point of the thread,
As I've said, logic is fallible when applied by any particular human at any point in time. A very common logical fallacy is to think that if you reject one part of a theory (like the existence of a god) then you must reject the entire theory (including all the lessons about morality that come with the religious package). If a human with faulty reasoning thinks they face a false choice of having to reject religion and morality both or adopt them both, then maybe they're better off choosing to be religious based on their intuition. The best counterexample of faulty reasoning leading to catastrophe is someone deciding that a loving god doesn't exist (they're more likely to arrive at this conclusion because of a bad life experience than from sitting down and reasoning it out, but regardless...) and therefore "rationally" (in their minds) deciding that the rules of morality that come along with their religion must also be rubbish.
So for individual humans prone to logical traps, intuition comes in handy from time to time.