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.
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")