The knight-in-shining-armour archetype hasn't had a decent run out since 1e's Cavalier. Paladin tries, but both the alignment restrictions and the religious aspect doom it. There's a gap here, and Cavalier fills it nicely.
Honestly, I feel the Cavalier is just a fighter. 5e is flexible enough to fill those niches with some sort of 'noble/aristocrat' background and 'mounted warrior/knight' archetype. The rest is just flavor and roleplay, and maybe filling out the build with some feats as you level up.
Barbarian - as I've been saying forever, it seems - shouldn't be a class at all. It should be a sub-race of Human, much lo=ike Wood Elves are a sub-race of Elf.
This is a good example as to why 'Barbarian' carries too much baggage to be a base class. Because in your mind, when you consider the concept of 'Barbarian', you think of it as a background, specifically an ethnic background, one which you find sufficiently different from 'High Men' to warrant it's own racial entry. And there is nothing particularly wrong with having a 'barbarian' racial package, or a 'barbarian' background package; but, if you strip off the class both it's racial assumptions (again, 'Norse' or some such) and its background assumptions ('primitive') there is still enough left to the core concept to be a class. You know this because you can still imagine characters who aren't 'chaotic', aren't 'primitive', aren't even racially distinct and yet still feel that character is best expressed as a Barbarian. For example, the Templars who have sworn to defend their deities temple and the clergy to their last breath, they may not have the skill at arms of a fighter, but they excel the fighter in will, commitment, and passion. They have the same fighting style and strength of a 'barbarian', even though they are civilized literate high men. The same is true of the handpicked body guard of the God-Emperor, each indistinguishable from the other, ready to give their lives in service to their liege. Likewise the elite shock troops of the King, nationalist zealots who hurl themselves against spears with shield and hammer, heedless of any danger, or the psychotic madman who prowls the streets of the city at night, slaking his bloodthirst induced maniacal rage. Likewise the goblin suicidal fanatics, in their drug induced battle-rage, are fundamentally the same sort of character. These all have the core abilities of the Barbarian, and in many published works you'll even see a nod of the head to this understanding, as characters who don't carry any of the baggage of being 'barbarians' yet are given the class because the abilities of a 'barbarian' like rage and indomitability suit the concept of the character. Proper class design wouldn't require us to ignore the baggage in order to make use of the class. Characters like the Norse beserker or the Aztec jaguar knight or the Maori warrior with their fear inducing and rage building Haka are just characters of this class who have particular backgrounds associated with primitive society. The class is much broader than that though.
Change the class name to "Nature Cleric" and see how those all look. Pretty much covers them all...
Not really, or not as 'cleric' is normally defined in D&D with its trappings of monotheism and its spell list drawn originally from the Bible, it's presumption of divine service and piety, and its relationship to the warrior-priests and crusaders of the middle ages. The animist with its pre-polytheistic viewpoint is the origin of the arcane tradition, but with much older roots than the Wizard with its loosely 19th century occult trappings, and self-referential 20th century 'fireballs' and the like. The Witch, the Vodoo priest, the Witch-Doctor, the Shaman, and truth be told the actual Finnish Bard are somewhere different than either the wizard or the cleric as it is typically represented in D&D. The Druid is an attempt at this sort of spell-list and flavor, but in my opinion one that carries too much specifics of Northern Europe as it imagined its own animist priesthoods of the distant past in the Renaissance and later and is as such ill-suited to the full range of characters you might encounter in myth and fiction. Worse, like the Ranger, it's implementation has become largely self-referential, forming an archetype of its own increasingly divorced from its original intention, so that it carries too many expectations in the reader. IMO, the best implementation of this concept in both mechanics and flavor was the 3.X Green Ronin Shaman, which is very different than a Druid (and in my game, killed the Druid and took his stuff).
The Aragorn self-sufficient hardy woodsman archetype, once the foundation of the Ranger class, has sadly been abandoned - to the point that now, ironically, it represents a big enough gap that a new class could fill it. I'd call that new class Ranger, and replace everything currently appearing under "Ranger" with it.
Or woodsman could just be a background. Aragorn IMO is a Paladin, right down to demonstrating the 'lay on hands' ability literally. Tolkien's "the hands of the King are the hands of a healer" has the exact same origin in medieval myth as the Paladin's ability to cure disease and cure wounds by laying on hands. And except for when we are introduced to him, when he was in disguise as a vagrant and woodsman, Aragorn spends most of the book as a mailed and mounted warrior. Indeed, if you read his backstory, he's spent most of his youth as a 'knight errant' and a 'black knight' in the courts of the Kings of the world. He's just a Paladin who also has some woodcraft, as a background or other skill investment. But it's a rather minor part of his character IMO.
I don't really know what you mean by filling the "self-sufficient hardy woodsman" archetype. I'm not sure what traits are so unique to that concept that it becomes a whole class on its own, but I suspect you could do a woodsman quite easily from what I call 'The Explorer' archetype, which would include not only the self-sufficient Wildman or wilderness hermit, but a huge array of other concepts as well, including for example a Pirate.
You could do this sort of thing pretty well in 0e-1e-2e, maybe even 3e. But 4e and 5e - particularly 4e - have such a huge gap between 'commoner' and '1st-level character' to make the everyman-hero archetype difficult if not impossible to reproduce. This one's not fixable by adding a class; it needs instead a few extra "levels" added in between commoner and 1st-level in order to work.
I'm not sure I understand you, but I'm sure you don't understand me. I mean this 'Folk Hero' or 'Ordinary Man Hero' to be a full 20th level class, balanced with other classes, and fully puissant as other classes at high level. I don't at all mean a 'commoner' although certainly heroes of common extraction are often heroes of this sort. Look at it this way - one way that The Fellowship is often looked at by gamers is that certain members like Merry and Pippin are but 1st level, while others like Gandalf or Aragorn or 'unimaginably high level' such as even 6th level! And there is definitely some merit to that viewpoint, but it wouldn't be much fun as a game because it's inherently unbalanced to have a party composed of 6th level characters that do the heavy lifting and 1st level characters that can't. So one other way to look at this is that the Hobbits were never low level at all, but were in fact 6th level 'Folk Heroes' and so - despite their superficial appearance of helplessness and lack of obvious prowess compared to Fighters, Paladins, and Wizards - were actually every bit as important and powerful as the rest of the party. Exactly how you capture that is another story, and I have some ideas that I think are pretty good, but it is certainly true that official D&D has never tried to make that an option.
While a valid concern, this one would be hard to implement and keep even remotely balanced at low level; as Sherlock-Holmes-like skills aren't the sort of things one learns quickly by adventuring but would have learned slowly and thus already had before 1st level.
On the contrary, the problem isn't making Sherlock Holmes over powered. The problem is making a merely extraordinary character like Sherlock-Holmes contribute amidst such extra-ordinarily powerful sword-swingers and spell-slingers. It's like trying to explain how The Batman is a full fledged and fully essential member of the Justice league when competing with characters like Superman, The Flash, Martian Manhunter, and Wonder Woman. Making 'The Adventuring Sage' or 'The Savant' or whatever we want to call him relevant is the hard part, and one I confess I've never remotely solved.
The other example of this kind of archetype that D&D simply does not do well is James Bond.
James Bond = Sherlock Holmes; fundamentally same sort of character, and in game terms same character class. In D&D though, a ridiculous array of skills and knowledge soon pales in value in most editions. This is why creating this character involves as much overhauling the way skills work as it does providing for the right sort of chargen. All those apparent high ability scores are, other than intelligence and perhaps charisma, actually just application of great skill. After all, at some point diplomacy is greater than charisma, and combat ability greater than strength.