Except they usually don't. Most class-based games are just as blank slatey as classless games, sometimes even more so.
Wrong. If you equate class with profession, and frankly this should be the least for a fantasy game, then it would define your skillset, know-how, lore, etc... yet this is already more than what any class-less system offers. Members of the same class can/should have differing skills, despite they share the same pool or basic skills. In a skill-based/class-less system everyone is literally the same "class" with little to huge differences in skills.
Example A: Imagine your party encounters a creature that sings just like a siren and someone must counter it/decipher the meaning, and decide what to do. In a class-based game, if you have a Bard e.g. in d&d games its kinda obvious whom to turn to. How do you decide in a skill-based/class-less system, if no one has a skill called "Decipher Magical Songs", only arbitrary magical skills like the ones in TES?
Example B: You have a class that is like a monk-masseuse-courtesan hybrid, lets call it Geisha. It is obvious they can be useful in quests related to back-pain and ball-dancing, and they can also do well in hand-to-hand combat and hidden weaponry, but how would you pinpoint this on a class-less system, in a ttrpg or otherwise? It is simply impossible, since if you have a bunch of finite few skills, you cannot possibly cover everything, whereas with classes you can make tidy boxes of concepts with which your players can maneuver in your game world.
In effect, by using classes you are enabling one of the most basic human behavior which is 'to categorize things and to put them into boxes', at the same time you also make it easier to design and to play your game (provided we are talking about a real RPG where role-playing and narrative matters).
In Baldur's Gate, you're always Gorion's Ward, doesn't matter which class you pick. Your background and your class have nothing at all to do with each other.
In Pillars of Eternity, you are... just some guy/girl who joined an expedition. The class you pick doesn't really have any proper ties into the world.
I don't get the differentiation here. I admit, BG is my favorite game yet I always hated to be Gorion's Ward since I hate the 'chosen one' cliché. But afaik in the travesty called PoE you are also a 'chosen one' only in this case some soul-reading reincarnating schizo, so what's the difference trope wise? In any case, this has nothing to do with class/classlessness since a class-less game can veritably use the same 'chosen one' cliché so this argument is pretty invalid imho and independent from our main issue here.
Picking a class doesn't make me feel like the character has a history in the world.
Optimally, it should, but let's just name how many things should matter (narratively speaking) in your average RPG:
- race/sub-race/nationality (if that's a thing in your universe)
- gender (and sexual habit if you're into that)
- class (or specific skills/skillset/power etc)
- faction(s) (which should depend on all the others, and there are mutual exclusions/hostilities ofc)
- alignment (I use personality-conviction instead, I hate d&d alignment)
So class is just one of many factors in an average RPG but it should be among the most important choices with the hugest impact on your gameplay.