The three levels Tao lists (roughly paraphrasable as novice - master - true master) can be applied to many other endeavors, e.g. art, martial arts, programming, as other point out in other comments. What I'd like to stress is the damage that's generally done in going from Level 1 to 2 in math teaching.
As a specific example take Analysis, generally taught using Rudin. I took this course in my graduate EE studies and detested it. I later thought about this and came to the conclusion that the main reason was the manner of exposition in Rudin where the classical approach is used: each chapter contains an endless sequence of lemmas, minor theorems, etc., one after the other with no discernible purpose and at the end of the chapter you get to prove a big result by using all that machinery. This approach, which dates back to at least Gauss (commonly attributed quote, which I couldn't find the source: "no self-respecting architect leaves the scaffolding in place after completing the building") not only is backward to the real course of events, it sucks the motivation by being so. At least it did so for me and for some other otherwise intelligent friends.
It's interesting that you say that. I'm trying to square this with my own experiences - I absolutely loved Rudin and to this day it's one of my favorite textbooks; but at the same time I recognize what you're saying about hiding the scaffolding and I share your distaste towards that.
I guess what I love in Rudin is that he gives the Level 2 details in such a lucid logical manner, with nothing missing and yet relatively tersely, all the details interlocking together. I got a sense of real beauty from reading him as an undergraduate that I did not get from other textbooks. I think that enjoyment ranks much higher from me than the disappointment from not getting the motivation early, which I do try to give myself when I teach something.
I wonder if these two can be separated: if a textbook could be Rudin-style in logical unity, terseness, and beauty and yet not "hide the scaffolding". To some degree, I thought Stephen Abbott's "Understanding Analysis" was a step in that direction, though it was still too wordy and meandering compared to Rudin, for me.
As a specific example take Analysis, generally taught using Rudin. I took this course in my graduate EE studies and detested it. I later thought about this and came to the conclusion that the main reason was the manner of exposition in Rudin where the classical approach is used: each chapter contains an endless sequence of lemmas, minor theorems, etc., one after the other with no discernible purpose and at the end of the chapter you get to prove a big result by using all that machinery. This approach, which dates back to at least Gauss (commonly attributed quote, which I couldn't find the source: "no self-respecting architect leaves the scaffolding in place after completing the building") not only is backward to the real course of events, it sucks the motivation by being so. At least it did so for me and for some other otherwise intelligent friends.