I do not know whether you've read part 3 of this series. It explicitly treats matrix multiplication as function composition, and goes with this at length.
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).
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).