Our culture enshrines rationality.
In my experience, enshrining something is the first step towards paying it mere lip service. I think our culture actually values rationality about as much as it genuinely understands it, which is oh so very slightly. At least, that's how I see it.
There is a term that I think encapsulates and values non-rational processes. And that's the movement that was a backlash against the Enlightenment Romanticism. Aesthetics, the arts, emotions - it's romantic.
The rationalism of the Enlightenment was a much needed corrective to the superstitions of former ages, but I agree it was taken too far when it came to deny in later stages the value of essentially non-rational experiences, etc.