"Beginning musicians play by ear, to the extent that they can play at all. Then they learn to read music. Eventually, maybe years later, they realize that music really is about what you hear and not what you see."
Reading music is also a way of learning a piece, and for complex music (i.e., most classical music) it is the most effective way of learning music quickly, even if you're the most advanced player in the world.
I would contend, however, that "they realize that music really is about what you hear and not what you see" is not just fluff. Intermediate to advanced players are frequently very tied to the page. They tend to think in notes and barlines, which is a reflection of intermediate level training that usually focuses on rendering literally what is on the page. One of the big advancements that musicians tend to make at conservatory is to learn how to think in audible units -- phrases and gestures -- rather than the visual units they see on the page. So yes, expert musicians do rely on their ear in a way that they usually do not at intermediate level. However, it also has nothing to do with the way untrained beginners use their ear, as a way of learning music.
EDIT: A more useful analogy for musicians would be to look at how they approach the physical aspects of playing. In the beginning, a string player is taught to move the bow by moving their arm back and forth. Once they get the hang of this, they spend a decade or so dissecting the fine details of that movement: fingers, wrist, elbow, shoulder. Making the bow change directions becomes a complex dance of many choreographed movements. Eventually, though, it all becomes second nature. When an expert thinks of moving their arm back and forth, it is a shorthand for all the things they've internalized about how to use the bow.
This is similar to the martial arts idea of learning technique and then forgetting it. It is most emphatically not, however, coming around to realize that your original conception of how to play was more right than what you were doing as a journeyman.