The real benefit of classes that isn't commonly mentioned in discussions is the flexibility it gives designers to create unique, coherent, and thematic mechanics packages.
Consider a simple example class like "Demon Sworn":
Ok let's see.
* At character creation, you choose a greater demon as your liege
Sure, why not. That sounds cool. But why does it have to be a specific class? Wouldn't a demon accept anyone's soul? A strong soldier's. A powerful wizard's. A devout cleric's would be even tastier.
Just have it be a character background or trait or whatever you wanna call it, like Fallout's jinxed or bloody mess, rather than making it a class. I don't see how choosing a demon lord as your liege determines which skills and abilities you can gain during levelups (which is what classes determine).
* Each month, you must make a minimum of blood sacrifices to your liege corresponding to your level
The idea of offering blood sacrifice to your liege is very cool. Dunno if I like the idea of making it correspond to your level, because level is a gamey abstraction of overall ability score of the character, rather than an actual in-world thing people have (people can see your strength by how muscled you are, but nobody can actually see what level you are). Still, good idea.
But I don't see how this is in any way class related. Any character can offer blood sacrifice. This is kinda like becoming a werewolf in Morrowind's Bloodmoon expansion. You have to kill and consume innocent NPCs on a regular basis to keep the hunger from fucking with you. It's a game mechanic that doesn't require you to be a specific class for it to work. It works no matter what the skillset of your character is, why tie it to class?
* For each sacrifice, you gain points to spend on gifts from your liege - such as demonic abilities, followers, treasures, information, knowledge, etc.
Again, why does that need to be tied to a specific class? In a classless system, the player could choose his rewards more freely. Wanna have a magical power or a powerful demonic weapon? The player can choose based on his build!
* Occasionally, your liege may send you on a quest to advance its agenda
Yeah. Again, absolutely no reason to class-restrict this.
* At level 20, you gain the attention of a demon prince, who'll send you on an epic quest to defeat your liege and claim its place as a greater demon
Why base it on level? Level is an abstraction, it's weird to tie such important story steps to levels. I'd much rather tie it to actual in-game achievements of the player (number of sacrifices performed, number of enemies killed, number of quests completed, ranks in influential factions gained, etc... something actually tangible).
How do you realize this in a skill-based system? The answer, more often than not, is you can't, because the core class mechanics aren't interchangeable.
What do any of the features you listed for that "class" have to do with class in the D&D sense? Wizard = can't use heavy armor, gets lower HP bonus on level up, can cast spells. Cleric = can't use edged weapons, gets medium HP bonus on level up, can cast divine spells. Fighter = can use all weapons and armor, gets a high HP bonus and attack bonus on level up, can't cast spells.
Demon Sworn = ?????
How does any of the stuff you described above determine which role the character plays mechanically in the game? Melee dude? Ranger dude? Magic dude? All of these are possible for your concept of the Demon Sworn.
It would absolutely work in a classless system. In fact, not one of the features you listed above even remotely imply class.
I would implement this as a background choice during character creation. Think Fallout's traits, except with more of an impact on the character. Maybe you can choose your character's background, the thing he did before he started the game, and that has effects on how the character plays and what opportunities he gets. Nobleman = he's got connections to other nobles and gets special dialogue choices with them; Sellsword = he once was part of a mercenary company that committed genocide in a small town and the survivors are now trying to get vengeance, leading to special random encounters on the world map; Prostitute = she gets a bonus to all seduction attempts but her low reputation means that the upper class doesn't want anything to do with her. Etc.
That doesn't turn these backgrounds into "classes". Backgrounds + classless system is in fact one of the best systems for roleplaying, much better than a rigid class system because it allows greater flexibility and therefore a greater amount of character types you can play.
A classless system allows you to, say, make a wizard character for whom you only raise combat skills on level ups, deciding to roleplay as a wizard's apprentice who got fed up with magic and decided to study combat instead. A rigid class-based system doesn't allow that. Scarcity of skill points means that he'll never be as good at either magic or fighting as a pure character, but he'll still be viable.