My idea - and it may not be right, it just sounds easy to comprehend...
I think it all comes down to how hard it is to "find" the tangent of the curve with a line that is 0.3 times the average gradient. If that were easy to find, we'd have picked that point long ago and been done with it.
That 0.3 Gradient point is the "real" best place to determine speed. It corresponds with the studies of people saying which print they liked best.
With everyone in the labs wanting a fixed point to look for, and where the curve meets 0.1 is easy to find, 0.1 became a popular point to look at.
So the ASA triangle was drawn up because when you hit the triangle - then there is a direct relationship between 0.1 (that is easy to find), and 0.3 Gradient (that you want but which is hard to find). So they made finding the point easy. In case there are philosophical disagreements between those who liked 0.1 and those who liked 0.3 Gradient... the argument is rendered moot because now you can look at it either way and get the same outcome. Actually a really good way to end an argument -- make everyone happy.
I think it also has something to do with being a good gradient to fit average subjects on paper.