Yes, I am aware that you can ADD story to a background. But the backgrounds don't DO anything. They can't level up, they can't grant new abilities. As a literary device they are nothing that hasn't already been in use as backstories. You can't tell new stories with that concept.
I agree there is a problem with the background, or at least there is a significant awkwardness.
Actually, a background can level up by means of gaining a higher Proficiency, such as +3 or +6. But as is, the only way to increase Proficiency is by increasing the level of the class.
One can level up by means of noncombat encounters. (In my games, noncombat encounters are worth the same as combat encounters, thus level at the same pace.) Hypothetically, a Farmer background can advance in levels by means of strictly noncombat encounters that relate to farming activities. Thus the Proficiency levels up with higher numbers, along with feats to supply Expertise, and perhaps more specialized features that relate to farming.
The problem with this use-only-the-background approach to leveling is, the lionshare of the character mechanics − the combat class − goes unused and is irrelevant.
To avoid having a huge irrelevant design space, is why it is better in 5e to use a statblock for noncombat character concepts.
Since the design space of each class level is roughly the equivalent of a feat, it is hypothetically possible to grant a feat at each character level, instead of a class level. There is still a character level, even without a class level. While advancing, the player can select noncombat feats relevant to Farmer. But this would be the same thing as creating a noncombat class, which would be unbalanced with any official 5e class in terms of combat. Thus the defacto noncombat class is nonviable for any official D&D 5e class, and for any typical D&D 5e game where the combat is central.
Nevertheless, noncombat is a technically viable way to play D&D 5e. But either most of the character sheet is the unused combat class, or the character is defacto a nonviable 5e class. Either way the approach of making a character strictly noncombat is problematic.
To avoid having a huge irrelevant design space, is why it is better in 5e to use a statblock. The statblock can list precisely what the character can do, without tripping over the gratiutous stuff that it wont or cant do. Moreover, it is hypothetically possible for a statblock to advance in levels − by means of completing noncombat encounters thus improving the Proficiency to higher numbers, and to other level appropriate statistics, plus accumulating feats for more statblock features.
There is no assumption that every statblock must be good at combat. Many statblocks are for a background-related concept, such as Farmer, Shopkeeper, and so on, who can lack combat capability. For the statblock mechanics, it is fine to lack combat. They dont need to balance, unlike 5e classes. The statblock and the character sheet are separate design spaces.
So let me try this again.
The idea from the overarching LitRPG storytelling model, is to take the concept of "class" and "levels" and "experience points" and "skills" and "abilities" and make them real, tangible parts of the world. As true in the fantasy world as the 3rd Law of Motion or Electro-magnetism is in this world. People know what their class is, they can look at a character sheet that breaks their existence down into numbers and abilities. And there are a lot of very popular and successful fantasy stories that use this approach.
For the litRPG genre, the story setting presents reality itself as some kind of roleplaying game, with game rules that the heroes of the story must master in order to succeed.
If you are familiar with it, the current anime tv series, Solo Leveling, is an example of the litRPG genre. Everyone in this world is a level zero character with negligible combat capacity. However, portals are opening up sporadically and fleetingly from an other world of monsters. These portals are imbuing the home world with magic that transforms a small percentage of the population, granting class levels. These are D&D-style combat classes, martial, arcane, etcetera. However, whether high tier or low tier, these people with a class level cant change their level because they are NPCs. The hero of the story is the only "player", the only one in the world who can advance in levels. He has gone from level zero to currently a high tier. (He estimates tier B, which seems about level 12 in D&D terms).
Especially for the litRPG genre, where reality itself is a game, there needs to be a clear and playable game. If the game is stupid or doesnt make sense, or doesnt have clear goals, then the game premise becomes a glaring distraction that detracts from any interesting narrative within the setting.
In other words, to suggest that a litRPG story doesnt need purposeful mechanics would be false. For the sake of the story, the actual game mechanics become more important than ever. In an actual D&D game, one can downplay, obfuscate, and reflavor mechanics. But in a litRPG story, the mechanics are front and center and cannot be fudged.
The same point as "I am a Maiar" or "I am the King of Gondor". Whatever use it has to the story and the world.
In a litRPG story, these identities like Maiar and King of Gondor EQUAL specific game mechanics. These rules must be coherent and part of a viable game that can actually be played. Typically, D&D 5e uses statblocks to describe these identities. At the same time, it is possible to write them up as a character sheet. For example, Gandolf approximates something like an Awsimar species and high tier Devotion Paladin, at least able to cast
Flamestrike, plus possibly a Folk Hero background. A litRPG story can explicitly refer to the Proficiency of a statblock (which corresponds to the tier), or to the actual mechanics of the Paladin class.
Sometimes, the litRPG story leaves the stats unspecified. For example, in Solo Leveling, there was an earlier encountered Japanese-style "goddess" (kami), who is clearly powerful but whose tier hasnt been revealed in the story yet. But certainly, this creature has levels and rules that the premise of the story revolves around, and that the viewer can speculate about.
You don't complain when people make up organizations like the Queensguard or the Dark Brotherhood. You don't complain when people say that all magic in the world comes from drinking the blood of a dead god, or from faerie dust. You don't complain when other fantasy concepts are used.
If Game of Thrones were a litRPG, one would be able to complain if their behavior contradicts the game rules of that are the premise of the story.
Moreover, one can easily translate the Game of Thrones characters into D&D 5e stats. I estimate most people in the story are level zero with a background only and without a level in a class. Many of the main characters seem the lowest tier, levels 1 thru 4. A few highest level characters seem more like the professional tier, levels 5 thru 8. Probably the Dragons themselves and the Night King are a higher tier.
I would use statblocks to represent most of these Game of Thrones characters, while assigning a Proficiency that is appropriate to their apparent tier.
Keep in mind, even the Dungeons & Dragons movie, Honor among Thieves, has official statblocks to represent the heroes of the story. Despite the setting itself referring to character classes, like Druid, Bard, and Paladin, these characters use the statblock mechanics instead of class levels.