The dichotomy is not that simple.
For example, in many areas of math it is of tremendous benefit to be able to internally visualize (with your "mind's eye") complicated structures and how they 'move' in relation to each other. Whether it's a skill or a ability, I've noticed some striking similarities to the internal visualizing skill needed in drawing:
To believably characterize a human figure in movement on a still paper that does not have ability to represent movement, it's helpful to see underneath the skin, to visualize all the muscles and bones and what they do when the figure moves. To draw a face, how the different muscles and bones and tissues are structured and form a face; to draw it with a certain emotion, how those certain muscles contract when you smile, and so on.
Or that was the gist of the my high school art class (I wasn't that good at it, but I have noticed the relationship.) And you do have a point it's useful to take a class (or find a book that adequately describes the contents of class). But the point of this example is that there's surprising amount of granularity [1] in the domain of mental skills, which is finer than "purely intellectual" vs "physical / emotional".
[1] in lieu of a better word
LW has a problem. Openly or covertly, many posts here promote the idea that a rational person ought to be able to self-improve on their own. Some of it comes from Eliezer's refusal to attend college (and Luke dropping out of his bachelors, etc). Some of it comes from our concept of rationality, that all agents can be approximated as perfect utility maximizers with a bunch of nonessential bugs. Some of it is due to our psychological makeup and introversion. Some of it comes from trying to tackle hard problems that aren't well understood anywhere else. And some of it is just the plain old meme of heroism and forging your own way.