The rules governing adjective order can seem technical and even esoteric, but there's a simple rule that is not in dispute: the determiner comes first. A determiner is strictly speaking not an adjective, but can look sufficiently like one that it is included in the "Royal Order of Adjectives":
The nine categories—in order from those farthest from the noun when
multiple adjectives are used to those closest to the noun—with
Determiner—articles (a, an, the), possessives (your, his, her, my,
their, our), number (ten, several, some), demonstratives (this, that,
In our case, Apple's is a possessive determiner, just like their or its. Indeed, you can substitute "their products" or "its products" for "Apple's products."
With that in mind, it's easy to see why "the most successfully sold Apple's product" doesn't work. You would never say "the most successfully sold their product," would you?
The substitution also makes clear why this suggested construction works:
- the most successful of Apple’s products
- the most successful of their products
With the phrase broken up by "of," the determiner once again comes first among the modifiers of its noun (there's only one modifier in this case, but if you were to add "post-1990," for example, you'd put it after the determiner.)
In the other suggested construction, Apple sheds the apostrophe s and, in doing so, goes from a determiner, which comes first, to an attributive noun, which comes last according to the Royal Order. Now "Apple" must always remain right in front of "product" no matter how you modify the noun. (See "bumper" in "the best-selling small political bumper sticker.") That is the case in our construction, so it is also correct.
- the most successful Apple product
There is no attributive noun form for their, so no substitution here.
Your editor friend left out one alternative construction, the simplest one in fact:
- Apple's most successful product
- their most successful product
See how the determiners come first?
Lastly, I should tell you that "most successfully sold" sounds awkward. I'd say "most successful" (as suggested) or "best-selling" instead, although this has nothing to with the grammatical question at hand. You can use your original adjectival phrase and everything I wrote here still applies.