Yes.
Heh, really?
So, only one class has Hit Points, and no other class has Hit Points (Strength-Fortitude)?
Only one class has AC, and no other class has AC (Dexterity-Reflex)?
Only one class can cast spells, and no other class can spellcast (Intelligence-Perception)?
Only one class can resist mental assaults, and no other class can (Charisma-Will)?
No. More like all but the class that gets the HP skill has the same hp, same for AC, mental resistance, etc.
As for spells, in such a system you wouldn't have a spellcasting skill, you would have a fireball skill, an invisibility skill, etc.
*A spellcasting skill would be the equivalent of having a warrioring skill.
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Only one class can use swords?
Only one class can use bows?
Only one class can wear armor?
Only one class can Stealth?
Only one class can know History?
Only one class knows how to Jump?
Same as above.
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Whether such a roleplaying game is even theoretically possible, the D&D roleplaying game has classes that share many mechanics in common with each other.
It's a starting point not an end point. It's meant to illustrate the possibility, not to be desirable. But we can take the starting point and then show any configuration of groupings is then possible. It's fairly trivial to do so from this point, because once we have all classes have unique skills of whatever arbitrary number and amount we want, then it's straightforward to realize that there's no specific grouping of such skills that is required, we can group them any way we want. So your str, dex, int, perception can easily become, swords, axes, maces, bows, hp, ac, damage resistance, fireball, magic missile, invisibility, history.
You don't actually need skills or attributes on anything you don't want the characters differentiated on. And some differentiations may work better via special ability overwriting the normal game procedure for that specific character than by skill or attribute bonuses anyway.
IMO. Most of the problems you cite with modern D&D stem from a few areas
1. Trying to be too granular.
The difference in a single attribute bonus barely matters in actual play.
2. Grouping specific skills to attributes (and really having that split between skills and attributes at all).
It's part of the reason a really intimidating Wizard is so hard to make in D&D.
3. Current system design makes dex, con and class main stat the primary combat stats and everything else if for out of combat.
This is fine except leveling in D&D typically come from defeating enemies and failure in D&D mainly stems primarily from dying in combat and so the combat stats are already inherently more important. If TPK's potentially depended on passing animal handling and medicine checks, then those would be top skills. If we had alternate paths relying on insight/intimidation/etc to defeat every enemy in the game and protect us from potential TPK then those skills would be just as valued.
4. The magic system
Magic does too much independent of everything but main stat (or i should say only dependent on class).
With balance in mind, I support player customizability, and a classless class.
I do too (well not the classless system). But Balance does not equate to 'no dump stats'. Those are 2 separate desires.
At the same time, I dont find "Atonal Music the RPG" a fruitful way forward for D&D traditions.
It's not so much that such a game would be Atonal, but it would just be missing a wide swath of potential tones. But again, the concept was a starting point to get to a particular place, not a stand alone example of a finalized game design.