Even the Cleric is just a D&D-ism that is barely retained anyway as Bards now have Arcane healing.
In 5e, Bards are neither traditional arcane or divine casters. From the 5e PHB:
Bards say that the multiverse was spoken into existence, that the words of the gods gave it shape, and that echoes of these primordial Words of Creation still resound throughout the cosmos . The music of bards is an attempt to snatch and harness those echoes, subtly woven into their spells and powers.
The bard spell list is actually pretty sparse in the PHB (some later supplements add a lot of other stuff to it, and therefore, a lot of power to the bard). Bards are intended to supplement these spares spells with access to
Magical Secrets at high levels, letting them grab a few spells from any other classes in the game with no ability to change your mind on this mini-multiclass power. In any event, this "words of creation" bit explains why they get access to such a unique set of spells, and why their spell list leaves off so much baseline.
Also while 5e bards and clerics are both full casters that are intended to use a weapon more than a wizard does, clerics have better access to offensive cantrips and at mid levels either get a boost to cantrip damage or their single attack a round. The bard has to actively pick a martial themed subclass to be able to really hold his own without expending spell slots.
Out of all the people who say "I just want the core three (or the core four) classes" the only time I've seen anyone create anything with that idea that I personally want to play or run is Kevin Crawford, creator of
Stars Without Number and
Worlds Without Number. He also doesn't just limit you to his core three classes (warrior, expert, and either psychic or mage, depending), he also gives you "partial" classes that split the difference. His splat content for classes are usually partial things as well.
Mostly when I hear that, I just walk on by, because someone advocating for a game with just fighter, mage, rogue, cleric is one paladin player away from learning why there's more than just four classes. The druid started as a subclass of cleric, but it sure works wonderfully once it's been established as its own class, as it can stop wasting space on things you can't do that a normal cleric can, like turn undead.
Hyperborea and
ACKS both offer up the four standard classes, then spend a decent amount of pages giving you other classes too. The obvious implication is that if you want to run the game and limit it to the basic classes, you can, and they are an intended and well designed part of the game (versus the power creep we saw historically, where the base clasess got sorta overwhelmed by later offerings).
But a game with just the four base classes, I just don't need. I'm not gonna run that game.