Warrior: Not-overtly-supernatural martial combatant built around combat maneuvers. Starts out more as an individually-focused damage-dealer but becomes more and more of a support/leader-of-armies type as it progresses into the later levels. Default fluff is, well, a pure martial - skill and strength developed through training and experience.
Hero: Overtly-supernatural martial with some combination of "always-on" buff abilities and active abilities that use power points as a resource. Stays focused more as a damage-dealer in combat, but gains access to preposterous out-of-combat effects at a comparable pace to spellcasters. Default fluff is basically a martial Sorcerer - innate, thematic powers, inherited by birth or by circumstance and refined through practiced use.
Rogue: Heavily skill-focused, with rules defining what having a high enough investment in a particular skill allows you to "just do" automatically. Primary combat role is mobile, disruptive support (vaguely akin to where the 5E Monk ended up), with Sneak Attack being a feature of some subclasses (Assassin, maybe Ranger) rather than a base class feature. Default fluff is much like the Warrior, but focused on out-of-combat skill rather than in-combat skill, leaning towards a general "on the fringes of mainstream society" vibe.
Mage: The generalist caster, works much like 5E fullcasters do: spell slots, lists of spells with discrete, defined effects, spell components, etc. Main difference is that spell lists are defined subclass-by-subclass, and are somewhat more narrow than any one 5E class's spell list; a Magical Secrets-esque ability to pinch a spell or two from other Mage subclass spell lists every few levels adds some customization, as well as the ability to learn any Mage spell they find a scroll for. Default fluff is basically the Wizard's core fluff - magic learned and perfected through study and tutelage.
Sorcerer: The specialist caster, with a small number of at-will abilities (a few from the base class, a few from the subclass), each with an extensive number of fluid modification options (again, a few from the base class, some more from the subclass) that are fueled by sorcerer points. (I'm given to understand this is basically how psionics have worked in previous editions.) Default fluff is an innate, internal affinity - whether from birth or gained by later circumstance - for a particular type of magic, refined through practiced use.
Channeler: The...variable caster, for lack of a better word - casts spells via skill rolls, with varied levels of effect depending on the result (plus some sort of mechanic to allow for scale-weighting). Potentially as effective at-will as a Sorcerer, potentially as diverse in potential options as a Mage, but not as consistent in effect as either. Magic for Channelers is always a negotiation on some level, in keeping with the default fluff - they draw their power from some source outside of themselves, be it a being, a place, or a concept.
Fighter -> Warrior, Rogue -> Rogue, Wizard -> Mage, Sorcerer -> Sorcerer, Warlock -> Channeler.
Most Clerics are Channelers, but any given Sorcerer subclass could reasonably represent a Cleric of the appropriate domain, and a Theurge Mage subclass would also work.
Druids could work as any of the three caster classes, and each would likely get at least one "nature magic" subclass. A dedicated Wild Shape user would either be a Sorcerer or a Hero.
Bards are split between Rogues and Mages as their most natural fit (one low-magic, one high-magic), but could also be represented as, say, a Song Sorcerer or a Myth Channeler.
Artificers are also split between Rogues and Mages - leaning Rogue - though this is probably the roughest fit as far as existing 5E classes are concerned; fortunately, it's considered "bonus points" anyway, so it's not that big a deal.
This lineup presupposes moving significantly more power (and somewhat more complexity) from class to subclass, leaving enough oomph for Warriors and Rogues to support half-caster subclasses, so each is an option for both Rangers and Paladins: non-caster Rangers are Rogues, supernatural-but-not-spellcasting Paladins are Heroes. More martially-inclined Channelers can also represent Paladins.
Warriors and Heroes would both have functional unarmed combat options and unarmored defense options by default, so both could reasonably support one or more dedicated Monk subclasses leaning into those options, while being able to use the core "unarmored martial artist" concept in conjunction with any number of other archetypes. Given the role overlap between my Rogue concept and the 5E Monk, the Rogue can likely support at least one Monk subclass as well.
Non-magical Barbarians are Warriors, magical Barbarians are Heroes.
As far as notable non-5E classes go: Warlords are Warriors (moreso at higher level, but ideally there'd be room to lean more in that direction from first level, especially in particular subclasses); Swordmages are Heroes (if non-spellcasters) or Warriors (if half-casters); Psions are Sorcerers but every class can reasonably support at least one psionic archetype; Witches work in every spellcaster class (though fit slightly better as Mages than the other two), or even as Rogues if leaning more into the alchemy and other out-of-combat magic.
If there's a class that could be trimmed for something else, it's definitely the Channeler, but in my view, the Sorcerer and the Warlock (/Priest) are markedly different core conceptual archetypes - internal versus external power - that lend themselves to very different base mechanics. The issue here, for me, is more in coming up with a mechanically-satisfying way to represent the channeling of external power, and to some extent not having another option in mind that cleanly replaces it: the other classes I'd want to use are mostly ones that the 5E PHB doesn't cover (Summoner, Shapeshifter, Scholar - the latter also taking some space from the Rogue and the Warrior), putting them mostly outside the scope of the challenge in the OP.