What was the original intended function of the core base classes? Honestly, to answer that question, we have to go way back to the mid 70s long before 3e since with the exception of the barbarian, bard, and sorcerer, all the classes go that far back.
For fighter and wizard, hell we're going all the way back to Chainmail, where the Fighting Man is a heroic unit equivalent to a whole squad of mooks and the Magic User is a human magical artillery piece.
For the most part, these classes had purposes that in many way were about flavor rather than crunch, and party role evolved organically.
The fighter was the combatant character. He gets the best attack rolls, has the most hp, and can freely use any weapon or armor found, and this is important when magical gear is found.
The wizard uses magic, but is terrible at combat.
The cleric originally began as a Van Helsing clone to combat a vapmire PC that wreaked havoc in Dave's early game, but it eventually got transformed into a fighting priest. He was more or less in between the fighter and wizard being able to fight and use magic but the other two classes were specialized in one of those areas.
The very roots of the LWQW problem are in place all the way back here, but the wizard was circumscribed by restrictions that keeps its power from overwhelming things. There are few limits onto how a fighter can fight, but wizards are frail, and have limited spell use. LWQW blew up like it did in 3e because the rules took a lot of the restrictions off. Of course the older editions looked at balance over the course of a campaign rather than sinlge encounters, and that skews perceptions too.
As for the other classes, they got added in to try to emulate characters from legends, myths and fiction. And again their roles tended to be more about flavor than fiction.
The paladin is the archetypical knight in shining armor, especially Arthur and his knights. The concept never really changed much. Unfortunately, paladin was the first serious hit to fighter, and very early in the game's history too being basically fighter+. But paladin was reigned in with very strict alignment and ability requirements.
Thief was probably inspired by characters like the Grey Mouser and other roguish fantasy characters.
The monk was inspired by a contemporary pop cultural interest in Eastern martial arts. Unfortunately, the class had a lot of abilities that didn't mesh well together.
The druid I think was the result of alignment restrictions on the original cleric. The cleric started out being a vaguely Christian warrior priest and was restricted to Lawful alignment. There were evil anti-clerics that basically served Chaos and worshipped demons. I think the druid started out as a neutral alternative to the cleric, serving the role as a sort of virtuous pagan. As clerics development and expanded to the nine alignments, druids stayed neutral and became a general nature priest.
The ranger basically stated out as an Aragorn clone.
These were all the classes that entered into 1e and then eventually made their way into 3e through 2e. The monk was the exception, 2e dumped them because it was felt they were out of place in a typical Eurocentric campaign which was the core standard pushed by 2e. Monk concepts were explored through several variants during the edition though.
As for the remaining classes:
The bard was built upon 2e's bard. The original bard was a messy proto prestige class in many ways and had a lot of inspiration from historical Celtic bards, but 2e turned it into a full class and made it into an archetypical magical minstrel.
Barbarians first appeared in UA, but the 3e barbarian was an amalgam of several different berserker concepts that were in 2e.
Sorcerers were the first unique class in 3e, and their intent was as an easier to play alternative to the wizard.
I’m sorry, but I don’t agree. Nine swords was NOT a good solution, falling into the trap of making everyone a Wizard. Simply removing the glass ceiling from martials and giving them better, but not necessarily Wuxia options was what people like me wanted, and the designers failed to produce that. There was a d20 RPG that did this, but unfortunarely, I’m drawing a blank on the name (I have it in storage, one of its features was momentum - I want to say it was named something like “Iron Age”)
I have to agree. I've only really skimmed over Nine Swords but the big problem with that book was that it didn't do its job. That job was to fix known problems with existing martial classes and instead the bbok just added more martial classes that use a quasi-magic system. That was absolutely not what the game needed. The base mechanics in core that were the root of LWQW were bad enough, but then the game compounded things by adding hybrid base classes that made the traditional martials obsolete. Nor did it help that 3.5 just loved adding more and more complex magic systems onto a rule set that didn't need more magic options and which had jettisoned a lot of the limits that had kept casters in check.