So the one thing I think is not quite correct here is that the TSR-era games (especially OD&D/1e) very much had the idea of protected areas within classes- what we refer to now as "niche protection."
While it is certainly true that there was often no significant backstory (in zero-to-hero, the character's story was created through play, instead of created through a narrative prior to play), the creation of a party was a group activity. We all know/remember that a party needed a cleric, but because of the way the game worked, you would need some amount of balance between classes- some amount of meat/muscle, some amount of healing, some amount of spellcasting, and some amount of thievery. Like the A-Team, you wanted a balance of classes.
We still see that today, but because the class-identity is so diffuse (no class is as weak in combat as the original MU, all classes have healing, any class can have stealth skills, traps are not omnipresent, most classes have access to spells (all classes if you consider feats and ancestry)) there is very little requirement of class diversity other than as a holdover from prior times.
I'm not sure we are saying things that are contradictory. Niche protection existed, to a degree, in 'classic' D&D, yes. Nobody is as good as the fighter at hitting things with a stick, nobody casts healing spells like a cleric, nobody blasts stuff and does everything like the wizard. Thief is a bit of a loser on that score, but they still have a pretty defined role and a couple of abilities that are hard to emulate (pick pockets for example).
And of course, it was perfectly feasible in every edition of D&D to create a 'team' through some sort of backstory. The originally conceived play structure of classic D&D, troupe play in particular, simply isn't that favorable to it. There's no formal backstory mechanism, PCs are largely fodder for the 'meat grinder' dungeon/wilderness, and the GM is often seen as an opponent. All of these discourage construction of some sort of unified party backstory or logic. Even if you strip away the troupe aspect and thus have a single fixed party roster, the other parts are against you.
I don't agree that modern D&D class identity is 'diffuse'. While 5e seems to me to cater to a bit of an 'everyone is a spellcaster' there are certainly strengths and weaknesses to each class. Anyway, if you look at 4e it has very strong role orientation and each role is quite distinct and useful. It may be that each class in modern D&D is not so fixed to a very specific set of abilities, but each CHARACTER is quite distinct (unless obviously the players choose to make nearly identical characters, something that any edition supports).
My point was more that modern D&D, because of your ability to choose your character options vs having some random die rolls do that, puts a lot more power in the hands of players to put their heads together and explain how they are an actual narrative team. The fact that you can usually expect to keep playing your character with a pretty reasonable probability, if you want to, also contributes. GMs are now seen more as facilitators of story than cunning exploiters of every character's weaknesses, so it is a lot less fraught to, say, have relatives. They may still be threatened, but now it is part of building a story vs the GM in the guise of some NPC trying to slaughter you.
So, finally, the type of concept like "I'm a ranger of the Order of the Arrow." is a bit more likely, more interesting, and more viable.