I don't think that intuitive knowledge is incompatible with science. A lot of the time, it's the intuition that forms the hypothesis in the first place. Like with the discovery of Penecilin. The scientist didn't have to think twice about the mold, but arguably it was his intuition that made him do so.
Of course, just like you say, once you get down to the grit of science intuition gets pushed aside. And I agree that this is a good thing.
But to say that intuition and science are incompatible is going a little too far.