I'll go ahead and add a 4th non-extant class that I feel should be a base class, and that's 'The Explorer'. Although The Explorer could be a specific example of 'The Truly Skilled/The Adventuring Sage', I feel The Explorer is a iconic figure that I feel is neither The Fighter nor The Rogue nor The Sage, but somewhere between them.
Nor the Ranger, I suppose?
D&D lacks a class that really fits well for a character that is defined by his ability to travel and move and negotiate a broad set of obstacles. IMO, 'Indiana Jones' is the iconic Explorer. Currently you can't even achieve the class IMO, and you can get close only by multi-classing.
Breadth of competency is something D&D seems to have trouble accepting. It has no trouble with tremendous versatility in spellcasting, on the theory that you can only have so many spells prepared, but simple, broad-based competence seems inconceivable in D&D class designs.
As I often have, I blame the original Greyhawk Thief for setting up that niche-protection/incompetence-generation precedent. Ironically, the Rogue outgrew all that and doesn't need the Thief niche protected, anymore, yet the Fighter still suffers a near-total lack of non-combat class features in respect for that and other vanished niches (and, less egregiously, Find Traps was nerfed into the ground).
Truly important archetypes ought to be achievable at 1st level.
Sub-classes get in the way of that ideal, certainly.
You can be a very determined and enthusiastic wizard, without the slightest either wanting to be an adventurer
Unless the world has colleges of magic and the determined wizard unlimited financial resources, the impetus for adventure is strong. If you have to go into tombs and ruins and onto other planes to dig up magical secrets rather than just go to Hogwarts, anyway.
much less (and this is the most critical point you are missing) wanting to be The Hero. It's very easy to play a Reluctant Hero right now in D&D.
It's very easy to play a Murder Hobo.
Technically, it's been a group of classes that had the metagame role 'Leader'.
Not really. The 'Leader' Role, quite specifically, was not necessarily a leader in that sense. (Nor was it 'meta-game.') You've made the point repeatedly that class is about the character's capabilities, not his role. In this case, the Warlord was the class that had the capabilities that allowed it to be a capable leader - or The Leader in the archetype sense. Whether it actually led the party in the sense of making decisions was entirely independent of those capabilities.
Outside of magic, such capabilities are virtually absent from 5e. You have a feat, a possibly-not-magical Bard feature (inextricably tied to a full-caster class), and a couple of very minor sub-class features (bordering on ribbons).
Fundamentally, I feel that 'The Warlord' is an archetype or subset of abilities that Fighters should be able to invest in. In other words, I think that as part of what makes Fighters balanced with spell-casting classes, they have to be much more broad than "hits things with a stick...hard". D&D has unfortunately got itself trapped in the idea that fighters are inherently big dumb brutes, and that's the limit of the archetype 'the fighter' covers.
The fighter could do with a great deal of expansion in competence outside of its current, narrow DPR specialization without running into any balance issues. It could be expanded to fill The Hero archetype quite handily, for instance.
The competencies of The Leader
archetype - and the Warlord class, which encompasses, but goes beyond the archetype - OTOH, go in a different direction.
The fact that leadership got associated with a terribly designed feat that was mainly trying to be backwards compatible with the concepts in 1e of achieving name level and getting followers is a big part of why we have terrible support for the concept of leadership.
I'm not sure it's about name level (you can take it at 4th), I think it may well have been a word game, though: "Inspiring" and "leadership" got thrown around a lot in the playtest, and 'Inspiration' got slapped on two different mechanics, and 'Inspiring Leader' on a feat.
There has been a lot of that from WotC over the years. A word gets used in complaints for a while, and they find a way to paste that word to something else. The most cynical example, IMHO, was 'Core.' For years, people played 'Core Only' to avoid all the broken crap in supplements (let alone 3pp crap), so, rev rolls, and suddenly every player-facing supplement has 'Core Rulebook' right on the cover.