I think intuitive thinking gets confused with 'touchy-feely' because 'it feels right' can mean two separate things.

With intuitive thinking, there is an "aha" moment, when a piece of the puzzle snaps into place and suddenly the picture or pattern makes sense, or begins to make sense. Or at least a hypothesis presents itself that one can find a way to test.

"It feels right" can also refer to reacting or making choices purely from emotion without considering what the emotion is actually signaling.

Intuition may 'feel' instinctive, but is not. Emotion and instinct are not synonymous but are closely related with respect to the basic emotions rooted in the limbic system.

Disclaimer here. I don't entirely know what I am talking about. I'm a psychotherapist, not a neuroscientist.

Anyway, while I think emotions and the capacity for empathy figure somehow into intuition (when working with people), and that intuition involves some unconscious processes that involve emotion, I think those unconscious processes also likely involve rational cognitive processes taking place 'behind the scene' so to speak. My understanding is people like Einstein, Hawking, and Dyson, brilliant theoretical scientists, are/were both intuitive and critical thinkers.