I agree, and here arises the forlorn expectation: we cannot display anything except displayable colors so let's not expect that. CMY (witholding K for the moment) is factually a 3-dimensional model. Increasing the quantity of a single ink subtracts more light of the wavelengths that it absorbs, and that variation is orthogonal to (independent of) the densities of the other two inks. This is the mathematical view and is not misleading. It does not include the non-linearity of ink combination, which of course should also be treated in the article.
 For visualizing a space-filling entity a picture is worth a thousand words, not least for the inexperienced reader. Flat drawings of 3-D objects in perspective are not universally recognizable and here we have the additional challenge of seeing through a filled volume. Please propose any better way. The article on Volume rendering will help.
coordinated adj. Organized, working together. I relate this to the mathematical or cartographic meaning of coordinate n. A number representing the position of a point along a line

You give reasons 1 to 4 for adding black ink all of which I agree should be in the article. The animation could omit black and I did look at that. However it makes a convenient arbitrary rotation axis which in my view stabilises the perception of the animation. It would also be misleading to show the colour space filled right to the black vertex without the help of black ink.

Since we are approaching this from mathematic/graphic and practical printing viewpoints respectively I wish to explore your recommendations for a new diagram. I guess you think of something like this and I shall be happy to look over any sketches which you may show here or send me by e-mail.