If your goal is real understanding, the approach that I stated is the right one. If you've learned it the other way and wish to actually understand it, you'll still have to learn the approach that I stated.
Your goal is explicitly not real understanding. And most textbooks do it in the order that you did. (Partly because professors think that most students aren't going to try to understand, so there is no point in giving explanation.) So you probably made the right choice.
But I would still like to see even a passing mention that matrices of just a representation of a linear function given two bases, and matrix multiplication is function composition. That could set a lightbulb on for someone struggling through a problem.
As for the rest, you might be right or not; but I am not competent to reform world's math education. If you can write better linear algebra guides than the standard textbooks, I'm all for it. Please inform me if you do that, I'd love to read that. (BTW It's not sarcasm - I'd really love to see linear algebra explained better than in standard textbooks. So far all fancy tutorials turned to be good only as entertainment).
Let me offer an abstract example. Consider the polynomials of degree at most 2. There is an obvious basis, namely 1, x, and x^2. A polynomial like x^5 - 3x + 2 can now easily be written in coordinates as (2, -3, 1).
However we have many other coordinate systems that might be convenient. For example suppose that we're sampling data, and can measure p(0), p(1) and p(2). How do we find what polynomial that is? Here is an easy way. We can easily find the new coordinates for our basis vectors: 1 -> (1, 1, 1), x -> (0, 1, 2), x^2 -> (0, 1, 4). That means that we can write down the matrix representing the identity transform (nothing happened), going from the basis we have, to the new coordinate system:
( 1 1 1)
( 1 2 3)
( 1 2 4)
( 2 -1 0)
(-3 2 -1)
( 1 -2 1)
There are a lot of problems where linear algebra comes up that you can think through more clearly if you think about things this way (complete with the role of the basis!) than if it isn't fully digested.
As for a better book, well, I already recommended Down With Determinants! :-)
Anyway, who is missing the role of the basis? It's in practically every textbook, including the one I used.
I agree with the maths that you've written, but it's the same thing that the textbook explains, practically in the same way. The road to understanding is not the same for all people, I guess...
There are now videos! http://www.linear.axler.net/LADRvideos.html
Maybe I'm hopelessly lost, but, for one, the inverse of
( 1 1 1)
( 1 2 3)
( 1 2 4)
( 2 -1 0)
(-3 2 -1)
( 1 -2 1)
As I commented in email, I did it by hand while very tired and made multiple mistakes.