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I'm ultimately conflating emotion and intuition because I don't think they're very different, and by extension, I don't think emotion and intuition are very different from "rationality." Whether to trust someone is a feeling you get from being around them; danger is one's sense of a situation. Likewise, the way we feel about things is dependent on what we know about them, which is to say, our past experiences with those things or things similar to them (and, for that matter, "similar to" is the same sort of sense about things).

This is probably better seen from a couple other angles. Probably the most interesting thing, since you bring up cognitive biases, is that confirmation bias can be the only way we approach information. If not for confirmation bias, we'd be incapable of talking about "signal" and "noise," or else we'd wonder, how do you tell the difference?

Or to go deeper, it's known at this point that formal systems aren't decidable in their application (i.e., the entscheidungsproblem), so it's just not possible that rationality is like a formal system. But then, what is it? My view is that it's just the same thing as intuition (the sense we have of things) but more of it, which goes under the charitable name "abductive reasoning."