classes spark character ideas and/or emulate stock characters or archetypes from the genre and/or make character building easier by presenting a complete/functional/contributing package of abilities without requiring undue system mastery.
The basic roles that speak to a well rounded party are still one warrior, one rogue, one priest, and one wizard.
If you run games that primarily focus on combat, that's true.
I don't agree with pickin_grinnin that the classic 4-class party is "well rounded" only in a combat-focused game. It also has certain strengths in a classic dungeon-exploration game.
But I do agree that it is "well rounded" only from the point of view of a certain sort of, fairly narrow, game play. Tony Vargas mentions archetypes and genre, and the classic 4-class party does not relate to archetype or genre in any serious way. The classic D&D wizard has its origins not in literary genre but in wargame mechanics, namely, artillery and related anti-personnel effects (eg Cloudkill is a fantasy equivalent to gas shells; Transmute Rock to Mud and Dig are fantasy equivalents to sapping and mining; etc).
The thief in D&D is not especially genre-based either, as it makes trap detection and lock-picking far more prominent than in most fantasy literature - these are skills that matter to a certain sort of dungeon exploration, but aren't a big part of the fantasy literary tradition. The weakness of classic thieves in combat is also not really genre-appropriate.
This clash between classes and literary genre is not an inherent RPG thing. It is a particular feature of the design origins of the D&D classes.
The Conan comics - particularly the long-running "Savage Sword of Conan" - do a much better job of reflecting the Conan from Howard's stories than the movies. Even so, the fact that Conan comes from a "barbaric tribe" and is physically imposing only represents a fraction of the traits of the character (as you obviously know). The only way you could really model him in D&D (from a technical standpoint) is with some careful multi-classing and some really good stats.
I think Conan is comparatively hard to model in classic D&D, because a fighter won't give him the requisite skills.
In 4e, I think a STR/DEX fighter specialising in melee AoE attacks and with either thief, ranger or warlord multi-class (depending on whether you're going for the early, middle or late-period Conan), plus some skill-training feats (for Stealth, Nature, Perception and Acrobatics adding to base Athletics, Intimidation and Streetwise) and the Unarmoured Defence feat, would do a pretty good job. He would be tough, fierce, and effective in a range of armours. An alternative pathway, which gets the full range of skills more quickly (4th level rather than 8th) but produces a less robust character who can't wear heavy armour effectively, would be a STR/DEX ranger.
Stats would be 16 STR and DEX, 12 WIS and CON, 10 CHA, 8 INT (Conan knows relatively little of magic, religion or - at least initially - history, which are the three INT skills in 4e).
I'm not sure what the best 5e pathway is, but I think fighter would be the base chassis.
Gandalf. If you look at how he (and the other wizards) are depicted throughout Tolkien's writing, it becomes obvious that he's semi-divine and falls somewhere in the lower orders of the pantheon of that world
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Merlin is more of a prophet or seer.
Although Gandalf in LotR is a type of angel, this is not a big part of his day-to-day capabilities. (And is not part of the character at all in The Hobbit.)
I think the real issues for building either Gandalf or Merlin in D&D are (i) that D&D wizards have their origins in artillery-style builds, and (ii) that D&D doesn't have very effective prophet/seer mechanics, and tends to lack influence mechanics short of outright control (5e's Charm is something of an exception).
Various types of bard build are one way to approach these characters, I think, where the more common magical effects become augmenters to ordinary activity rather than the very frequent casting of overt spells.
The heroes in fiction can always hit and kill their enemies when the writer says, and this is typically instant or almost as quick.
The way that combat is resolved in D&D is an artefact of that particular system, not RPGs per se. 4e used minions to generate the literary trope of the quick kill, and it is possible to have combat systems that work on a single check (like skill checks in 3E or 5e), just as it is possible to have non-combat resolution that requires multiple checks (like skill challenges in 4e).
I don't think D&D is ideal for emulating literary-style fantasy combat, but I don't think it has to be hopeless.