The current amount is good.
I go back and forth on whether we need a Psion that isn't an Aberrant Mind Sorcerer (that uses Spell Points instead of slots and Int instead of Cha), or need a Shaman (that isn't just a Warlock with a Primal Spirit pact), but I'm happy and can build pretty much everything from this set.
Unlike some folks I often agree with here on the Enworld forums, I think it's a great thing that 5e has both big tent classes like Fighter and Wizard and small tent classes like Paladin or Warlock.
I think there's a gamist role for the 3 Gish classes that aren't just subclasses of Fighter or Rogue a la Eldritch Knight and Arcane Trickster, allowing for a more balanced and integrated focus of magic-wielding warriors in the game. But by definition, such "dual class" classes are more narrowly defined that the broad tent classes they're "children" of. These "Gish" classes are akin to what have been considered Prestige or Promoted classes in other editions or mediums, and thus in some media can get away with having both at once. But they're often lackluster in their storytelling precisely because Paladins are so often just "Warrior/Knight+Priest/White Mage." To make sense of them, finding core mechanics that make them tick and can unite a bunch of different archetypal ideas that are all Gish but approach it differently is key.
As you might guess, I'm a fan of the Paladin, (Revised) Ranger, and Artificer in 5e. ALL THREE have found said hooks that allow for a bunch of different archetypes, even if less broad than their parent classes, while still united in their identities as Gish.
IMHO, the reason Ranger was so lackluster in the PHB was that it lacked those key unifying working mechanics. Deft Explorer, Favored Foe, and Primal Awareness change all that, in the same way that Lay on Hands, Smite, and Auras work for Paladins, and Magical Tinkering, Infuse Item, and the Right Tool For the Job do for Artificers. These ideas enforce the fiction of Resolute Oathsworn Gish, Open World Adventure Game Gish, and Techno-Hero Gish.
Artificers don't feel like Bladesingers or Hexblades or Eldritch Knights, but Armourers Forge Adepts, Battle Smiths, and Artillerists very clearly fill that Martial+Arcane box while Alchemists and Mavericks feel right at home with the concept while showing what they can be without being on the front lines.
Rangers don't feel like Fighter+Druid, and Paladins don't feel like Fighter+Cleric, in a large part because they've got these host of other abilities that Fighters and Clerics and Druids don't get. But they have shared abilities too, between shared cantrips and shared spells, and shared "ideas" about magic, and that's critical.
I don't think reducing the game's class list down would be helpful. It would lead to more "balance" of concepts, but I don't think the concepts need equality. What they need is to be balanced gameplaywise so that players aren't screwed over. And they need to carve out their own niches to represent the supporting fiction and new ideas in ways that aren't interchangeable.
There are a handful of ideas in the supporting fiction and past editions that still aren't possible in 5e. I would like to see those added, and MAYBE there's a class out that could be added in some capacity, but I really see most everything being possible as subclasses going forward.
I agree. There is no official arcane half-casting class in 5e, but there are a few subclasses that try (and fail) to fill that role.
Artificers say hello.