See Brett Victor’s 2011 proposal: http://worrydream.com/KillMath/
I met a friend many years later who sadly was still forced to do that rref()-by-hand for even larger systems of equations in university! That left no time to actually learn anything useful in linear algebra. Madness.
https://theodoregray.com/BrainRot/ has some nice ranting about this (though it does go a bit off the rails when it starts talking about video games).
Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?
The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.
i.e. |u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2).
This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.
That said, as someone who uses notation for math regularly, I want to keep using notation. It’s a helpful tool, and it is an efficient and precise language.