I'd say Greyhawk is more pulp fantasy than sword & sorcery, given Gygax's broad scope of influences. You've got tons of faux-medievalism in the nation structure, tons of weird elements on the fringes, and whatever else he thought it fun to include. However, Gygax eschewed meta-plots with Big Bads threatening the world (ala Sauron) and favored a tone of picaresque adventurers looking to make their fortunes. In that there's a strong S&S vibe to his adventures and campaigns, if not his world.
For a classic Greyhawk vibe, I'd say any ruleset should:
1. Emphasize play in the first two tiers (up to 10-11 level) to keep things away from demi-god level. Use much slower advancement, particularly past level 10.
2. Advancement dependent on gp as xp as the default to keep gameplay suitably greedy and grubby.
3. Emphasize standard races, particularly humans, over more exotic races (drow, tieflings, dragonborn, shardminds, genasi, etc.) or monstrous humanoids. GH was very much about normal people going out into the Big Dangerous Weirdness that surrounded civilization. If such were to be included, I'd consider giving them an experience penalty and certainly disadvantage on charisma rolls when dealing with 'normal folk'.
4. Possibly a more robust henchmen and hireling system with recommendations to quickly resolve the efforts of your retinue.
5. The rest of the 'rules' could remain as is or use the standard DMG variants (slower healing) as desired. They are all close enough to GH to work as a modern translation, thereof.
6. Then, you'd have DM advice for creating GH-styled adventures and campaigns, with a focus on handing out treasures and magic items, incorporating weird aspects and challenges (Hint: "A wizard did it."), and so on.
The biggest schism between D&D and true S&S is the role and nature of magic. In D&D, magic is a known, quantifiable tool. In S&S it is shrouded in mystery with its practitioners mad, tainted and/or inscrutable. People need to sacrifice, often everything they hold dear, to ascend to the heights of magical power. Magic is portrayed as unnatural and corrupting and contrasted with the clean, natural power of flesh and steel. This is the Conan/Hyperborean style of S&S.
To create a suitable system for 5E, you have to go much farther than a GH-inspired set. We should probably:
1. Ditch non-humans from the race list. You CAN make a S&S world with the D&D/Tolkien races, but that flavor is a lot to overcome. Best to just ditch it. In place of race, I'd consider expanding background to the weight of a race (with ability bonuses, improved features and such) to give diversity. Creating nations/human subraces is possible, but gets icky fast. Best to just focus on backgrounds.
2. Remove all spellcasting classes and subclasses. Sure, magic-users exist in these worlds, but generally as NPCs and, in particular, villains.
3. Create a spellcaster class or perhaps subclass of rogue. This class would mostly use skills and natural abilities, but dabble in magic (rituals and low-level spells with a few slots). Spell selection would be mostly defensive and utilitarian (nothing flashy). Possibly add a casting roll with consequences on failure (perhaps 1 level of fatigue per spell level). Dabblers can learn other spells, such as attack or higher level magic, but these are considered black magic which require sacrifice to learn (possibly a reduction in max hp or stats or something stranger) and have much greater consequences for failure (fatigue, damage and/or it backfires horribly).
4. Definitely emphasize feats as a way to customize and add back in some power to the PCs. Probably add more feats and perhaps allow PCs to learn additional feats as rewards.
5. Once again, emphasize lower tier play (1-11 or so) with slower advancement. I'd say advancement should be based on either treasure gained or, ideally, some system of new experiences (steal from a sorcerer's tower, lead a pirate crew, kill a dinosaur, etc.) to keep the PCs suitable vigorous.
6. Some guidance on using player agency and, then, skills as the core resolution mechanics in lieu of magic. Guidelines for players to get into the S&S-mindset and understand the genre conventions.
7. Guidelines for the DM to create S&S worlds and content.
Finally, with have the Gonzo/Cosmic Sword & Sorcery fantasy of Elric and similar works. I'd submit that 5E can model most of this, at least in tone, with its available ruleset. The deeds of Elric are mostly indistinguishable from a high-level D&D character and, indeed, if you want to run high-level D&D, the Eternal Champion stuff is nearly required reading. Mechanical changes and other considerations are:
1. For a Young Kingdoms-type world I'd say standard non-human races are probably fine. Melniboneans are basically elves, for example. I'd put them in small or secluded enclaves keeping human predominant. In a high-level Eternal Champions-type game weirder races such as tiefling or dragonborn from other dimensions would certainly work.
2. Classes, as is.
3. For a lower-level Young Kingdoms game I would, once again, emphasize lower-level play with slower advancement.
4. For a high-level Eternal Champions game I would actually level-lock it, allow the PCs to pick select appropriate magic items (their Stormbringers and what nots), and then summon them by Fate to fight the forces of Chaos (or whatever) in some mist-shrouded hell. Obviously, without advancement, this sort of thing probably works best as a one-shot or a series of recurring, sporadic adventures rather than a long-term campaign. The enjoyment, however, is not in the leveling, it is in the playing.