Older settings never worked that way. PC races, classes etc where added to the game in Dragon Magazine, Unearthed Arcana etc, and they became part of the setting.
Yes, it is strange to see so many people taking a purist approach that tries to freeze these old kitchen sink settings in amber.
In the very early years of D&D it was assumed that any DM worth their salt would homebrew everything. Even after TSR started to publish official settings and modules, it was still understood that DMs would customize things so that no two Greyhawks would ever be exactly alike. I had both the gold GH box set and the grey FR box set, and I never got the impression that anything in them was set in stone. They were clearly sandboxes or tool kits that the DM could use as a starting point for their own campaign, and eventually high level PCs might change things even more. There were also huge blank areas on the map marked “here be dragons”. The DM could always introduce new elements as something that just arrived from the other side of the world, or emerged from an obscure dungeon.
Real old school campaigns often included all kinds of wild gonzo stuff drawn from everywhere, so I don’t see why new classes or races from later editions present such a problem. As early as 1976, EGG himself crashed a spaceship full of robots and laser guns into Greyhawk! Tieflings in particular seem like they already belong there anyway. Gygax clearly loved infernal intrigues (a taste I never shared), as can be seen in all those entries for archdevils and demon lords in the monster books, in the modules he wrote, and in those terrible Gord novels. There is even a whole country named after the half-demon who tries to rule it. Just rename them “alu-demons” or “cambions” and there you go.
I think things began to go awry as FR introduced ever more convoluted lore in an avalanche of products that fans were expected to keep up with, and got much worse when they decided to use metaplot to introduce rule changes for new editions. I did not play during the 3E and 4E eras, but I would occasionally flip through D&D books in stores or browse the wikis, and stuff like the Spellplague or the Blood War did not impress me at all or make me want to start playing or buying again.
So many fandoms and franchises these days seem to be consumed by an obsession with canon consistency: superhero comics & movies, Dr. Who, Trek, Wars, etc. It can become suffocating and prevent anyone from doing anything new. D&D in particular does not need this as DM and player creativity is supposed to be the whole point of the game.