I want to focus here for a second. Because this I think is a fundamental problem with your approach to this question.
The question is "What if everyone in the setting had a [Class]?" And your answer is "Not everyone in the setting would have a class, classes are special."
You are fundamentally altering the premise of the question.
Classes are something "special", namely combat competency.
In my current settings, "not everyone would have a class". Not everyone is competent in combat.
And yes, you would get a positive response from those in the thread who do not like the idea of a setting where everyone has a class, but you are not actually answering the question.
I answer the question. Even if in my current settings not everyone is competent in combat, many are. Many cultures train for combat, including warriors of a clan, and militia of town. It is easy to imagine where everyone in a setting has at least some combat training, hence levels in a class.
And secondly, your approach of "just give them backgrounds" fundamentally creates a massive amount of tension. Because you would have some people with a "class" that grants them these abilities, something they can point to. And other people, people who are usually far weaker, not having a "class" but something else that does not give all the benefits of the "class".
Well, yes. A character without a class is "weaker" in combat.
But the noncombatant still has skills, toolsets, proficiency, and feats that they can point to with regard to noncombat areas of life.
And since you can be born with a class, which is a necessity for Sorcerers to be a thing, then you have some people who are born more special and more powerful, in a demonstrably objective way.
Even the Sorcerer class needs training and experience to advance from background to level 1 to higher levels.
Nothing assumes that innate magic must be for combat. Perhaps the sorcerous magic is strictly for noncombat and weaving clothing (Weavers Toolset, Performance Skill).
The weaponization of sorcerous magic is combat training and experience.
One of the fun things, as a person who enjoys stories where classes are a real concept within the world, is not seeing what a level 20 fighter can do, but what a level 20 Farmer or Barkeeper can do. In asking the question "what does it look like to be a practical demigod in THIS field or THIS specialty." But by separating them out and declaring "anyone who isn't doing combat and going to combat schools to get real classes can't do that" you are taking out a big element of what makes the exercise worthwhile in the first place.
There is such thing as a Farmer background whose Nature skill and so on eventually advance to proficiency +6 (and expertise +12).
However, I dislike the design concept of a "Farmer class". A high priority for the design of any D&D class is to be balanced compared to other classes thus equally effective in combat. Unless weirdly weaponizing the concept of a Farmer, the hypothetical Farmer class would be strictly inferior in combat, thus no longer be a viable player class, or even a "class" at all.
Or, the DnD rules don't accurately represent the totality of reality. Only the combat section.
The question, to me, is aimed far more at "building a world with classes" and far less at "make DnD combat classes a real thing" because if everyone has to be good at fighting, the world strains far more than if you allow non-combat classes. Which DnD has had in the past and make sense for building this sort of world.
D&D evolved from a combat game, and a "class" represents a combat style, and doesnt represent the totality of reality.
At the same time, there is no need to translate everything into some kind class. There are other design spaces to describe noncombat things, especially background, proficiency, and feats. Where each class level is worth about a feat, it is hypothetically possible to build a character without combat features, without a class, who only selects noncombat feats. However, at this point, the character would be unsuitable for D&D adventures that require combat. Also, it becomes more straightforward to use a statblock instead of a character sheet to represent the character.