I just looked up mnemonic major. Neat!
Much of that order isn\’t surprising. It matches the mouth and tongue shape and position.
P and B have the same mouth position, the only difference is P is plosive and B is voiced. In Gregg, voiced has a longer line. In Pitman, voiced has a thicker line. (Reading hint: If a line length isn\’t clear, try a sound mid-way between plosive and voiced, or just try one then the other.)
M and N use the P and T mouth positions, respectively, but are also nasals.
The vowels also use mouth shape and tongue positions. The circles are back vowels, the hooks are front vowels. The small-circle sounds are higher tongue than large-circle. (Roughly.) And yes, I really is ah-ee. In singing, you stay on ah for most of the note, and switch to ee at the very last instant.
(My son took speech therapy, and my mid-life crisis included singing lessons. I find it fascinating how phonetic theory shows up in so many different places.)