I wouldn't contest the idea that PS is its own thing.
I WOULD contest the idea that Greyhawk ISN'T its own thing. Greyhawk is UR-D&D. Trap-filled dungeons, mad wizards, Balance Wins, fuedal wars, a strong "Sword & Sorcery in a Deadly World" vibe. Greyhawk is where you go to be as old-school D&D as D&D can be.
FR isn't like that. FR is history and epic timescales and legendary people and massive magic. It takes Greyhawk and adds a touch of legend and mythology. If Greyhawk is grimly deadly and Vancian, FR is Tokeinish and super-magical.
Dragonlance isn't like that. Dragonlance is naked heroism, a narrative setting for good vs. evil struggle with wacky sidekicks, Dragon Opera if you will. It's plucky-heroes-against-a-great-evil.
They are these things just as Planescape is shades-of-grey stories about extreme beliefs cramming up against each other in a grand philosophical conflict.
Which is to say, not excluslvely, and not always as played, but these are the things that make these settings stand out from each other. If I wanted to play a pure dungeon survival high-mortality game, I wouldn't pick Planescape for it, I'd go play Greyhawk. If I wanted a game about exploring ancient ruins and uncovering lost history to overcome present evil, I wouldn't doink around in Dark Sun, I'd dive into FR. If I wanted to play a heavily narrative game that was like Star Wars but with dragons instead of space ships, I'd bang right into Dragonlance. FR is not the most comfortable fit there. Meanwhile, if I wanted to play a game with shades of gray and with sympathetic villains, I'm looking at PS first. All of these are certainly possible in any setting, but some settings are going to be better fits than others by the nature of what those.
I mean, if I want to play a game about the struggle of divine order against primordial chaos, 4e's Nentir Vale is perfect for the job. If I try to tell that story in PS, the players are primed by the setting to ask all sorts of uncomfortable questions I don't want to consider -- "well, maybe the rampaging engine of fiery destruction has a point" is a great angle for a PS player to consider, but it's not something I want to muck about with in that kind of game. Go Nentir Vale, use the 4e alignments, and there's no question: Chaotic Evil engines of destruction are objectively worse for everyone than even Evil corruption of souls. I clearly CAN tell that story in PS, but it's not the most comfortable fit.
And all of these games are D&D games.
So I challenge the notion of "generic." Dungeoncrawling through a mad wizard's tomb isn't any more generic than "all halflings worship Yondalla." It's a specific expression of what D&D can be. It's a primordial expression, but being closer to the origin gives you no special claim to "default" in today's diverse fantasy ecosystem. It's where the game started, but anyone entering the game today can pick from a limitless menagerie of fantasy worlds and tones to embody. They should pick the one that best suits their goals for the game (or make their own if that suits their goals the best!).
D&D settings are tools you can use to enhance the feel of a particular kind of game, packages of places and NPC's and Proper Nouns that reinforce certain ideas about the desired experience you are having.
That's true with 5e's implied setting as much as it was true about Nentir or Greyhawk or Planescape or Dark Sun or your homebrew setting.