[Inspired by the comments from Thoughts from a Sandwich.]
There's no question that the universe is "organized" in the sense that it's very highly likely (although not certain) that the universe is not just a collection of unconnected, random events. Nobody (besides a few radically skeptical philosophers) disputes this conclusion.
The real question is: what is the best explanation for this organization? Intuition is fine in its place, but coming up with the best explanation requires more, much more, than just intuition.
Our intuition is part of our natural cognitive apparatus, our brains viewed at an abstract level as our minds. It is the result of our past, biologically and socially evolved ways of thinking about the world, which worked under the circumstances our ancestors lived in to solve the problems that affected their survival and reproduction.
We can have some confidence in the reliability of an intuition, but only under the circumstances under which the intuition evolved and for which the intuition was subjected to selected pressure. Whenever we extend our intuition past those circumstances, we can have no confidence whatsoever that our intuition actually applies reliably to the expanded, non-historical circumstances.
We appear to have obtained the ability to consciously think scientifically by accident. Indeed the ability to consciously think at arbitrarily high levels of abstraction appears to be accidental. Conscious scientific thought is a mere five hundred years old, far too little time for biological selection pressures to have made any impact on our physical brains and time for social selection to have made only very little impact on our traditional, learned ideas.
We find something very interesting when we double-check our intuition using scientific thought: Under those circumstances where we see very strong selection pressure for accurate intuitions, the result of our intuition matches the result of scientific inquiry. For example, our intuitions about the macroscopic properties of ordinary objects (mass and size; rocks and trees) matches very closely our scientific understanding of those properties. There is no metaphysical a priori reason for this correspondence; there's nothing in the definition of scientific thought that logically entails it must match any particular intuition. The correspondence is a posteriori, "after the fact". The a posteriori correspondence goes both ways; science specifically gains credibility precisely because it does match our reliably predictive day-to-day intuitions about macroscopic things.
When science fails to correspond to our intuitions, it is always the case that our intuitive predictions of what we should see fails to match what we actually do see. This "failure mode" is a priori: it follows from the definition of science, which takes what we do in fact see as an authoritative epistemic foundation.
When taken out of their original context, our intuition becomes radically unreliable. We don't need to compare our intuition against scientific thought; we need only observe: our intuition leads us to expect to see one thing, and we in fact see something radically different. Counter-intuitive findings of science are all over the place; Lewis Wolpert has written a whole book on such examples, The Unnatural Nature of Science. (Wolpert uses "unnatural" in the sense of "counter-intuitive", not "supernatural".)
And this is what I think Sagan means when he says, "But I try not to think with my gut. If I'm serious about understanding the world, thinking with anything besides my brain, as tempting as that might be, is likely to get me into trouble." [thanks, Dagood] When taken out of its original context, it is misleading to rely solely on an intuition; we must employ the more rigorous methods of conscious science. We can employ our intuition for conjectures, but we cannot rely on our intuition to just hand us the whole truth, or even in many cases anything close to the truth.