(Edit: Added the last sentence of the first paragraph, to make my point a little clearer than it was. Nothing has been taken out or otherwise changed.)

I doubt there’s a direct connection. Possibly Gregg knew about the Major system, but he needn’t have; Gregg Shorthand wasn’t the first such system to use these groupings, and the first may not have come from Major either. They likely all share structures because they are trying to solve similar problems—reducing the English language to a small number of naturally ordered symbols—even though shorthand and mnemonic systems ultimately have different purposes.

Almost all phonetic systems of shorthand share this structure, because these are the consonant sounds of most English dialects. (Some consonants, like “ch” and “j”, are not simple sounds but combinations of two others. They’re just frequent enough to take their own shapes.)

The voiced and unvoiced variants (t and d, s and z, etc.) are naturally grouped together in both Gregg and the Major system because they’re cognates. The “S” and “Z” sounds are produced in the same fashion when spoken, except the vocal chords vibrate when making a “Z” but don’t with an “S.”

It is interesting that neither Gregg nor the Major system distinguish between voiced “th” (as in “this”) and unvoiced “th” (as in “thing”), but many shorthand systems do the same for the sake of simplicity. We don’t distinguish between these sounds in written English, so why introduce a new complication for learners of shorthand (or the Major system)?

(Gregg also doesn’t distinguish between “S” and “Z,” but that’s only natural as well, as the sound “Z” is almost always written with an s in English. See: days, means, rims, does, fends, reason, etc.)

If you look through other shorthand systems, you’ll find much the same groupings. Pitman’s Phonography came before Gregg; even little-known systems that came after it, like this one out of Pittsburgh, did the same.

It would be interesting if the Major system grouped vowels in the same odd way that Gregg does, as Gregg doesn’t precisely follow English phonology there. (For instance, having the long and short cognate vowel sounds in “grate” and “bet” separated from each other.) But it looks like there was little use for vowels in the Major system, except for the semi-vowels which are commonly called consonants.

Thank you for pointing out the Major system to me, however—I hadn’t seen it before and will read into it!