I've always felt that the 'most general magic-user' role is filled by the wizard, as evidenced by their immensely broad spell selection.
Well, right. That would be the paradigm shift. Instead of "Wizard" being the base magic-using guy, the <renamed/replacement Sorcerer> becomes that base...with potential access to the wizard/arcane spell list, the cleric/divine spell list, the druid/nature spell list, whatever else the guy adds/includes. So you can still have your divinely flavored Favored Soul "Sorcerer", your "Storm Sorcerer" that has all/any arcane and druidic spells that involve weather, your dragon sorcerer's whose magic all relates to their parentage's element/abilities, elemental sorcerers, shadowcaster sorcerers, etc..., etc...
I don't fancy the notion of a base class that exists just so you can cast even moar variety of spells.
Well, that's what I'm saying...they wouldn't be getting "moar variety of spells." They'd be getting a specific list, according to their type. Think of it as, instead of
adding "bonus spells" to the existing sorcerer's spells, your sorcerer is getting access to
ONLY the "bonus"(themed/flavor) spells. Which, then, they can play with in terms of metamagic, doing different things with them via spell points, and a bit of utility stuff built into features, cast innately/without "study" [though I have no problem with the story/fluff of studying/books in my magic and see them rather as basic/necessary as well as historically, mythologically, and literarily relevant], and so on like that.
The reason we, of course, think of the "Wizard" as the base/general magic-user is because that's what they are/have always been.
Magic-User --> Mage --> Wizard/Sorcerer --> add in Warlock -->Wizard/Sorcerer/Warlock.
Since the Wizard used the spellbook/memorization mechanics that we know & love, that is the obvious successor/extension to what a "D&D Magic-user" was. That is the paradigm shift that would need to be overcome...and much easier to do without the name and preconceived baggage of the "Sorcerer" class.
What we think of as "Sorcerer", with their spell points and metamagic and innate magic to weave into existence to various effects becomes the base "Magic-User [Caster]." The Magic worker who doesn't have to do any work or adhere to any specific flavor/story/casting mechanic.
Then, as the original idea was put forth, "Magic-User/Caster + study/spellbooks/researched preparation" flavor & mechanics remains what we know and love as "Wizard."
MU/Caster + worships the gods/religious/support style magic flavor & mechanics remains what we know and love as "Cleric."
MU/Caster + worships nature/shapeshifting/elemental magic flavor & mechanics remains what we know and love as "Druid."
MU/Caster + extradimensional non-god patron/invocations/short recharge magic flavor & mechanics remains what we know and...like somewhat...as "Warlock."
MU/Caster + primordial music-magic/charm/inspiration/primordial music-magic flavor & mechanics remains what we know and...don't mind so much but prefer with druidic spells...as "Bard."
It's not so much changing anything about the classes we have, other than re-envisioning and [gods help me] re-"branding" the Sorcerer, but changing their position (and making them a legitimate class/giving them a reason to exist in the process) in a hypothetical "class tree/order of things" sort of way.
I also am pretty set against any new base classes that aren't impossible to approximate using the existing tools (class, subclass, feat and prestige class).
But that's my opinion, and you know what they say about opinions.
Yeah, I'm with ya on that too. This could certainly be done via sub-class, as I said. You just have to homebrew out the base class spellcasting and replace it with the flavor spells of choice.
For myself, since I don't use the existing sorcerer anyway, rewriting the whole thing for homebrew use is no real big.
So then, you could have <Renamed Sorcerer/Innate Caster> base class...and then the same kind of thematic origin subclass break up we have now...just a lot more granular in flavor (en par with the Clerics/Wizards we have).
You could go the opposite way, and try to follow the kind of thing they've sort of based the existing Sorcerer class around. Namely, instead of having to have a "Draconic/Elemental/Divine/Chaos/Storm/etc...etc..." origin, you'd go more generic and could have subclasses that are simply broad strokes: "Ancestral" for your sorcerers that have magic in their blood because of lineage, "Circumstantial [Fated is probably a better name]" for got splashed with chaos goo, absorbed gamma radiation, hand picked by some deity/natural battery for divine energies, "Savant" for your parents were making whoopey during a lightning storm, the person -or maybe the very magic of the world- has an affinity for a specific elemental type or specific spell school (illusion only, transmutation only, etc...), etc...