To be clear (and clearly I wasn't!

), I'm not advocating for the 9 square grid coverage of each role and power source being covered... or that some square isn't covered twice or covered in different ways from different classes.
(Also, it's also likely not an actual 9 square grid, but that's a very strong element in architectural theory/history/design, so I went with it...

)
Rather, what I'm looking at is how, from 1e onward, certain classes have a thematic/profession flair to them, with abilities to support it, while others do not. Look at the fighter in 2e -- the whole class fits on less than a page, and a good chunk of that is the followers table for what happens at level 9. There's nothing else to them. Yet the Ranger (my go-to example for this) gets bits on tracking, on interacting with animals, on creature hunting, and so on. Druids get a whole thing on their organization and ethos.
Because of that, it is both limiting but also can get confusing and narratively weird. What if you want to have a band of wilderness scouts who protect the villages and the forest? Do they all need to be the Ranger class to be a Ranger? Why couldn't a veteran cleric in that group be as adept at survival and tracking as the "Ranger" Ranger?
I like the capabilities that come with groups like the Rangers or Druids (and the flavour too, though that needs to be left so that the DM can customize it for their world/campaign), and don't want all classes stripped down with nothing to replace them. Which is what 4e PHB did, in many ways -- there's almost no Ranger-esque riders to the Ranger class anymore, instead the class is really detailing a skirmishing dual-wielder or archer.
So my suggestion, which at first might seem counterintuitive, is to go full bore and strip every class down so they are all focused on their powers (and rename ones like the Ranger that have historical identities), AND also very much add another layer to cover that second layer of capabilities and flavour, such as being a Ranger, as a profession/calling. Thus, you'd have something like your Power/Class (HP, HS, Defenses, Weapon/Armour proficiency, some abilities, and your AEDU-ish powers), and your Profession (Skills, additional non-encounter based abilities, perhaps some additional U powers on a separate track).
To that I would also follow the now-common idea to split up what's traditionally covered in Race into Parentage/Ancestry (no ability score modifiers, but a few flavorful abilities (like dragonbreath) tied to the physiology), and your Past (which including culture and backgrounds, skills and perhaps an ability).
Then one can include Theme, Paragon Paths, and Epic Destinies.
But back to classes, yes, very much I agree we do not need to hit every combo of every type, and only once. With this redesign the resulting list of classes would hit their intended marks really well while being even more evocative/flavorful in how they use their power sources. And I think there's an advantage to adding on the extra layer of a profession/role in the world, separate from their role in the party/encounter.
I'm...not really sure how that responds to what I said though.
By stripping out "Fighter," "Paladin," "Swordmage," "Warden" etc. and replacing them with the Defender role and your choice of Source, you are necessarily eliminating the mechanical distinctiveness of each class. "Defender" now means one (and
only one) set of mechanics. Likewise, "Martial" means one (and
only one) set of mechanics, regardless of whether you are a Controller, Defender, Leader, or Striker. This is definitional to the approach:
every Striker is, and must be, a Striker in exactly the same way, because the Striker mechanics are totally source-independent. There can be no distinction between the
way that a Barbarian or an Avenger or a Sorcerer do their Striker thing, because there is one (and
only one) way to be a Striker. That way might have toggles like ranged vs melee or something, but ultimately it has to be identical in what Striker options it grants, because you could always have chosen a different power source.
This isn't just about the bells and whistles of the experience, the wilderness survival element of Rangers or the entertaining performance element of Bards. It's the
whole package. The thing that makes a "Martial Leader" (formerly Warlord) a
Leader is, and must be, exactly the same thing or set/range of things as what makes an "Arcane Leader" and "Divine Leader" (formerly Bard and Cleric) a Leader in this system. All mechanical differences between classes of the same role vanish,
must vanish, other than what is directly provided by choice of Source. You can't have unique Mark punishments or uniquely party-friendly AoEs or a distinction between "tanky bruiser" damage-dealing and "mobile skirmisher" damage-dealing, because everyone chooses from a perfectly identical list. That's the whole
point of making it "choose from the list of four roles, and also from the list of five sources. That determines what character you play."
Your proposal to add a new category, Profession, is sort of a halfway effort at what I described as remaking the old system but with more steps. That is, you had already understood (before my post, I mean) that this approach strips out identity and leaves something same-y, and you wanted to respond to the idea that just because "Ranger" (to use your canned example)
does fulfill the "wilderness survivalist" class fantasy, doesn't mean that has to be the
only way to fulfill that class fantasy. Which is fair! The problem is, again, you would be doing so by needing to make these things
generic.
Like, let's look at actual 4e Themes here, because I think they are illustrative. I
like Themes as they are, and like you I would want to see them expanded into a full on "Heroic Origin" mechanic that scoops up culture, personal life history, and all the things that get a character
started on their Hero's Journey. But the problem with your proposal—of making it so anyone wanting "wilderness survival" goes to the one-stop shop of Theme-town—is that that "Wilderness Survivor" theme must be
separate from everything else their character
is. You can't have any connections between
any combination of Role, Source, or Theme, because the player could just as easily decide to do any one of them differently. Each of the combinations must be valid, and the whole point of this redesign is to make it simple and elegant without the (allegedly) crufty wastefulness of having a distinct Martial Striker who Does Wilderness Stuff, and instead having, "Pick your Role, Source, and Theme, or roll 1d4, 1d6, and 1d100 on the following tables." (Getting to 6 sources by having Martial, Divine, Primal, Arcane, Shadow or Elemental, and Psionic.) Making "Shadow Striker" be a Striker in a unique way, different from how a Martial Striker is a Striker, would be obviously counterproductive to the goal of simplifying and streamlining, but that's exactly what you would have to do to retain the mechanical distinctiveness of Rangers vs Assassins. Having a Theme of "Hired Killer" vs "Ex-Commando" doesn't, in any way, bring back that "we are both Strikers but in very different ways" element; it simply offers a way (itself entirely generic) to support the narrative and as the name implies
thematic elements.
To reiterate, I
like the fact that Themes, and
only Themes, are this generic. If they are generic but classes are
not generic, we get the best of both worlds: each class can be mechanically distinct and tailored to a particular set of class fantasies, while still supporting players who want something else or who don't want some of the baggage that comes with a particular class. That hypothetical "Hired Killer" Theme lets
anyone get some of the flavor and mechanics of being an Assassin without actually needing to be that class, so they can be a ruthless and cold-blooded Wizard or a no-nonsense mercenary Fighter or whatever else tickles their fancy. The "Ex-Commando" lets
anyone be a grizzled veteran or special forces operative, whether they're a Cleric from a frontier parish or a Bard who's had to rough it between gigs or whatever else. Ideally, there would be
many Themes, or flexibility in exactly how each is implemented, so that this can be used to narrow in pretty specifically on what the player wants to play.
This way, you can still have the cool differences between Barbarians and Sorcerers and Rogues, while still getting the "not everyone needs to be a Ranger in order to be a wilderness survivalist, and indeed even the Ranger itself doesn't
need to lean all that hard into it." You can still have Bards with the unique ability to multiclass freely and Wizards who are the absolute best at Ritual Casting and Druids that have shape changing powers, without having to force players who want to be musical performers or learned scholars or mystics to specifically be Bards, Wizards, and Druids respectively.
I will say, as an aside, you did leave out rather an important aspect of the Ranger that
is actually wilderness-y: the option to have an Animal Companion. Which was actually reasonably balanced, unlike 5e where it sucked and took them like six or seven
years to fix. That's a bit part of respecting and remembering the "wilderness survivor" element of the class without
forcing people to take it if they just want to be a badass melee or ranged weapon specialist. (There's
also the Fey Beast Tamer Theme, though, for folks who want a fuzzy buddy but don't necessarily want to be a
Ranger to get one. Shaman and Druid also enabled some pet-related options, meaning you could have a variety of choices on that front, and all of them were at least tangentially related to Primal magic or the Fey.)