In other words, you should draw a circle not because someone told you to draw a circle, but because the thing in front of you genuinely has an underlying circular form at that size and in that position -- as Ames must have done when he came up with his drawing books in the first place. And the more you do this, the faster you learn the underlying structure of things, which makes it easier to draw from memory, draw from imagination, or simply draw from life without needing to stop as frequently to measure things.
Which isn't much of a problem for things like still lifes, but it becomes a bigger problem in, for example, portraiture, where the human eye is particularly sensitive to errors in proportion -- which is why Edwards includes some construction rules-of-thumb in her portraiture chapter.
You may be focusing too strongly on "circles and squares" as idealized shapes... If you look at the picture of the elephant in the article, none of the shapes is quite a circle. Rather, they outline masses, like the blobs you might see if you defocus your eyes. Being able to capture the proportions and locations of these masses is actually very important when drawing from life. (see, for example, Nicolaides's "The Natural Way to Draw")
It's been a while since I've done any drawing, but from what I remember you can easily use both in the same image - blocking out the constituent shapes with approximations with around the right proportions, and then making sure that the negative space looks about right in the gaps in between.