To say good chess players (not to say grandmasters, let alone world champions) don't calculate moves borders on insulting.
For instance, I know that if I have two apples, and give one away, I now have one apple. I also know that the prime factors of 87 are 3 and 29. One of those I "calculated" in my head, the other I know intuitively. Of course for both my brain performed calculations, but thinking of it like that is rather missing the point.
Anyway, I don't think that we disagree on the actual point here, that claiming that somebody has "intuition" isn't somehow attributing their proficiency to the occult.
Intuition (whatever that is) plays a great role, especially at the top level.
I vaguely paraphrased this from a paperback chess guide of mine (hopefully more credible than the internet). According to my guide, grand masters can effortlessly memorize the positions of realistic matches. But for "nonsensical chess matches" where the positions were randomly generated, the masters were just as bad at recall as the noobs. From this data, my guide inferred that masters rely on recalling attack-patterns from their experience more than brute-force calculation. Similar to how a tourney-strength player has openings memorized by heart, I imagine that a master recognizes common middle-game patterns like the back of his hand.