Really? People aren't that variable IRL. Like genuinely. There was some vague loose notion in the early editions that ability score modifiers correspond to standard deviations. 68% of people have |z|<1; 95% of people have |z|<2. Even though that's only on a single score, going with the 95% figure, .95^6 = 73.5% of people lack even a single ability score modifier bigger than +/- 1. 10% of people have no nonzero modifiers of any kind. The vast majority of people are, in fact, very minimally variable.
Sure. I'm fine with this. Most of the bell curve's population is in what I call the mushy middle; and the question is how to game-mechanically deal with the extremes on either end of the curve.
The problem in game terms is that the WotC editions assume linear rather than bell-curve progression when applying bonuses, giving the game-mechanical difference between 10 and 12 in any stat the same mechanical heft as the difference between 18 and 20, or between 3 and 5. This is where things fall apart, as the difference between 10 and 12 (or even 8 and 12) shoud be mechanically zero as you point out, with bonuses and penalties only accruing at the outer ends of the bell curve.
It is an artificial, genuinely unrealistic expectation that characters should have highly variable stats--doubly so when we start factoring in a dangerous world that has razor-thin margins of error, as is typical in early-edition D&D play. Under those conditions, it's actually quite unrealistic to expect that some Fighters have a strength of 6 and some have a strength of 18/00 and some have 14 etc.
That's why in 1e there were minimum stat requirements in order to be a class. A Fighter had to have Strength at least (9? I forget and can't be bothered to look it up right now). A 5 or lower in a stat dictated your class; two stats of 5 or lower made the character unplayable as it couldn't qualify for any class.
We not only can, but should expect that extremes, especially low extremes, should be quite rare--because few people who have such shoddy strength would do all three of (a) bother trying to be a Fighter in the first place, (b) stick with being a Fighter through all the training where they failed and failed and failed and failed and (etc.), and (c) never got any better at Strength things as a result of their training.
We shouldn't expect someone with Str 3 to be a Fighter but that character could very well be playable as a mage type, or a talky type, where physical strength isn't a requirement.
Like...failing at something a lot of times but never getting even the slightest bit better at it is kind of crazy, and reflects either a profoundly damaged human being (since most characters in early-edition D&D are human) or someone incapable of growth and adaptation...which means they shouldn't survive their first dungeon, let alone their tenth.
Wouldn't your hypothetical Str-3 Fighter (which even in my game can't exist in practice) put level-granted ASIs into Strength?
Not sure, though, how we got to a Str-3 Fighter in the first place.
My point here is just...people only get as variable as the dice indicate when you look at the extremes. And when you do look at the extremes, you're necessarily going to see extremes that...work with what they have, or that got better at what they sucked at. Which means most Fighters are going to be fairly strong (or dextrous, if that's their bag). Most priests are going to have a little wisdom, even if some are lower than others. Etc.
I think we might be agreeing here only from very different directions.
I also think people tend to improve in what they practice at, and it makes sense that members of most adventuring classes are going to practice at those things which improve what they do as that class: Strength for a warrior type, Intelligence for a mage type, etc. The way to reflect this improvement-through-practice in 5e game mechanics would be to force level-based ASIs to go into those "primary stats".
Personally, I'm of the opinion that players should be given an incentive to have crappy numbers in "good" stats like Dex, Con, or Wis. That way, players actually want to opt in for "these stats make my character succeed less, but experience more interesting things".
Before doing that I think the class-agnostic elements of the different stats need balancing, though I've no real idea how to do this. Dex is way too powerful (and has been in every edition), Con is too powerful (though less so than in the early days), while in the early editions Cha was fairly useless and often dumped.