I get what you're saying...but is this really as "either/or" as that? Does it have to be one way or the other? Why can't a class based system have both?
Well, let's put it this way. A system can have both. But if it does have both, I'll feel the need to rewrite it to clean it up.
But in particular, given the nature of mundane versus magical, dividing up magic-users into increasingly narrow classes that can't do everything might make sense provided you were willing to similarly gimp the spell lists of iconic concepts like Wizard or Cleric. That would create classes that just weren't equipped to direct the story in every possible situation. But dividing the fighter up just simply does not make sense because mundane is an inherently smaller set of ways to drive the narrative than magic is.
Right. But why can't a class-based system say: if you want to be Robin Hood, you could be a Fighter who specializes in Bows/Archery or a Ranger who does the same?
You certainly can and certainly D&D 3.X did and was a highly successful system. But ultimately, part of what brought 3.X down was the crushing weight of 'there is more than one way to do things'. Some freedom in how you create an archer is fine. In my game you might make a fighter, a hunter, a champion with the Hunt portfolio, or an elf of just about any class. Or you might mix it up and take a little from several areas. Heck, the archer in my current group is a multiclassed Explorer/Sorcerer who uses divination spells to guide his arrows Jedi knight style. But I think I've been careful to distinguish between them. The fighter is distinguished by what he fights with - bow. The hunter is distinguished by what he fights against - say undead. A priestly elven archer would still be foremost a cleric that happens to fight with a bow. My explorer/sorcerer has a very different feel and overall set of skills and strengths than an equivalent fighter specialized in archery would.
On the other hand, I can think of almost no differences between the concept of an aristocratic, honorable fighter specialized in mounted combat and a cavalier. I can't think of anything a cavalier 'ought' to be able to do, that our hypothetical fighter couldn't do. Creating one from a fighter in my game in a very archetypal fashion would look something like:
a) Assign you abilities in a balanced fashion and take either take 'Unusual Background (Aristocrat)' as your starting trait to broaden your skill list to include things like Appraise, Diplomacy, and Knowledge (Heraldry & Nobility) or else take Noble Rank if you want to start play as an actual knight.
b) Take a lawful alignment. Write into your backstory something about honor and chivalry. Preferably align yourself with some noble clan, temple, or higher ranking aristocrat. Figure out your station in life - a penniless but admired Knight of the Road, a landless third son seeking a leige, a down on your luck mercenary, a household knight, a last survivor or disgraced outcast, a noble bastard, or a rakish gallant seeking adventure.
c) Take the Courage feat with your general feat (or bonus human feat).
d) Spend your fighter bonus feats on mounted combat. Max out your ride skill.
e) Spend some of your skill points on Leadership and/or Tactics. Put whatever is left into things that fit your background.
f) Keep doing that as you level up, and as the mounted combat feats near capstone gradually start taking more feats to enhance your ability to boost and protect allies in combat.
Now, if I created an actual Cavalier class at most it would just be a fighter with the equivalent of a class ability equal to a bonus starting trait and the equivalent of a smaller and less flexible bonus feat list. The Cavalier and the Fighter of the same level, at any chosen level, would look almost identical and have essentially the identical stats and concept.
Where you seem to want to go with base classes is exactly the opposite of where I want to go. You seem to be happy with the concept of every one could have basically the same character, but of different classes. What I want from base classes is everyone could play the same class, but each have radically different character concepts. Thus, what I want is that you could have a party of 6 Fighters, or 6 Fanatics (think Barbarian), or 6 Hunters (think Ranger), or 6 Rogues and they'd all be so distinctive they leave plenty of space to develop your character without feeling like the middle child in a big family.