Warriors:
--- Knight (shiny armour, etc.)
--- Swashbuckler (light-armoured, ranged or melee)
--- Veteran (basic mercenary type)
--- Ranger (woods warriors, non-casters, no animal companion)
I'm not seeing all that much difference between the Knight and the Veteran. The difference in reality was one of money and social standing (you had to be able to afford your armor, weapons, and horse), and there's nothing preventing a Veteran from fighting on horseback or a Knight from fighting on the ground.
This list is definitely forgetting a few thing. There's no ranged-weapon specialist (archers and knife-throwers are quite different from Swashbucklers) or Brute/Brawler (artless smashing or trained punching).
The only reason why Rangers shouldn't start with an animal companion is that animal companion should be an ability available to anyone. There's no reason why a fighter shouldn't have a loyal dog or horse, or a thief can't have a trained monkey, or a cleric can't be accompanied by an animal considered sacred to their faith.
Rogues:
--- Thief (basic sneaky-scouty type)
--- Assassin (hired killer or spy)
--- Charmer (talky persuasive type)
Anyone with a high enough Charisma can be a talky persuasive type. The only reason it's commonly associated with rogues is because media often has "charming rogues," because if they
weren't charming, it would be too obvious they're actually @$$holes.
--- Dancer (genericized wire-fu type, replaces Monk)
This... doesn't make much sense. There's more to "wire-fu" than moving around a lot.
Casters:
--- Cleric (divine caster as usual)
--- Nature Cleric (a.k.a. Druid, covers Shaman as well)
I have to say that if you think Druids should be kicked because they're part of one particular culture, then Shamans should be as well. It
also refers specifically to a single group (indigenous Siberians) and some people think it's been misappropriated--and it certainly
doesn't mean the "savage tribal magic user" that a lot of people envision it as and that AD&D has historically used it to mean (along with Witch Doctor--a concept invented in England to refer to English people who protected themselves and others from English witches during the 16th century). If anything, shamans are clerics who have the ability to communicate with and command various spirits, including ancestral and divine spirits, not just nature spirits. The name shaman also certainly isn't a catch-all term--according to Wikipedia, Native Americans do
not like that term being used with them. If a 6e were to have such a class (which would be cool), then it should probably be called something like Spirit-Caller or Spirit-Talker,
and the concept of the spirit world should be more fully fleshed out (and not just relegated to being a type of fey),
and an educated cultural consultant and sensitivity reader should be employed.
--- War Cleric (covers Paladin also)
I was in a game once that had a paladin of Chauntea--the Realms'
agriculture goddess. Considering the sheer number of homebrew cleric domains you can find, there's a large call for clerics that aren't just "kinda paladins" and healers. Same for paladins; the idea of the holy warrior isn't just for deus vulting infidels.
--- Diviner (divnination spells etc. mostly go here)
--- Illusionist (mind-screwers, charmers)
--- Necromancer (they make things dead then play with the corpses)
--- Summoner (summoning, gating, conjuring spells and familiars/animal companions exclusively go here)
--- Mage (artillerists and buffers)
Oddballs:
--- Psionicist (if it can be made to fit without hosing Illusionists' screw-with-your-mind niche, debatable)
--- Tactician (maybe best as a non-adventuring class?)
Why make them a non-adventuring class? For that matter, it wouldn't be hard to give a tactician adventuring abilities.
--- Artificer (or Tinkerer; specifically for games/settings that have lots of devices and tech)
If
anything should be a non-adventuring class, it should be the Artificer or Alchemist types, since it should take days, months, or years to make the things they make, not rounds or a long rest.
What's intentionally missing?
--- Swordmage - the classic "I wanna do it all" class - gone in any form.
--- Bard - can't make it fit between Charmer and Illusionist - gone.
--- Warlord or Leader - classic "I'm the boss" class, also healing is strictly Clerical.
--- Barbarian - can't make it fit between the other Warrior classes except as a very boring one-trick pony - gone.
--- Multi-classing - gone.
I
vaguely agree with Bard not for the reason you posted. I have zero problem with a "dabbler" class--a little bit of fighting, a little bit of magic, a little bit of thievery. I feel D&D doesn't do a good job with the idea that magic takes on different forms for different magical traditions (they
talk about the idea at times, but don't show it mechanically). For instance, every class uses the same components, even sorcerers, who theoretically should at most need something to help them focus but not require verbal or somatic components. And they definitely don't take cultural differences into consideration. Hence, the reason I vaguely agree with Bard is that all the magic being the same
except that Bardic magic is also musical seems weird.
Musician can be a background.
(The ideal solution would be to take in-setting cultural differences into consideration.)
However, Level Up gives Bards "Battle Hymns" (in addition to full spellcasting, which I feel is a bit too much), which is one way to give Bards magic that is both useful and thematic
and doesn't step on anyone's toes, and such a concept could easily be expanded to give them a wider range of abilities.
That being said, thematically, Bards are different from Illusionists, since their magic is (or should be, IMO) more about communication, diplomacy, and entertainment than it is about messing with other people's minds and senses. And the Charmer isn't needed because anyone can take that role.
As for the others:
I'm not personally a fan of swordmages but I don't see a problem with them. They can do
two things, not "all," and they tend to not be not nearly as good or versatile as full fighters or full casters.
Warlords/Leaders: No, magical healing shouldn't be strictly cleric. Or if it is, it should be strictly the province of clerics of healing, life, fertility, and similar things. There's little reason why a war god or smith god should have healing clerics, after all. Also, this is that old "are hit points actually meat points?" argument again. If hit points aren't purely meat points, then there's no reason why a Warlord--or Bard, even--can't restore some; they're not healing wounds, they're increasing morale and giving the listeners a stronger desire to survive the battle, and restoring hit points is how that's represented.
If anything, limit magical healing and make nonmagical healing more effective; healing potions, at least the weaker type, should be more common, cheaper, and easier to make by people with knowledge of herbalism or alchemy. If we can assume magic is real, then we can also assume that there are plenty of semi-magical plants and other materials that can be cultivated and used. Let people with herbalism spend some time in the forest and then boil up a healthful tisane.
Barbarian: While I'm also not a fan, I think that
many people would say that they are not even remotely boring. However, they should
not be "savage tribal warriors," since a berserker concept is definitely not limited to that idea. It's a type of warrior that can be said to enter a very violent fugue state. If you have no problem with a presumably dispassionate assassin who kills for money--equally "one trick"--the barbarian shouldn't be a problem. They could also be turned into a semi-magical fighter; instead of spells, they channel magical forces, elemental/natural (storms, earthquakes, infernos, or simply bestial forces), otherworldly (madness from demonic GOO, or chaotic sources), or even divine, like the Greek Maenads.
That would give them a niche.
And no to getting rid of multi-classing.
Many people like to fully envision their characters before they make them, and it's not always going to be in a way that completely maps to a single class. And lots of other people find that play takes their character in different directions to what they imagined. I didn't start Rime out thinking she'd be as much of a front-line fighter as she ended up being, but once her role became obvious, it made sense to multiclass into fighter.