To quote a beloved sci-fi character, "I am nothing if not a product of my upbringing."
I started playing in 1998: the end of days for TSR, back when there were two versions of the game still (sort of) in print, oD&D and AD&D2. Both of these games defaulted to the 3d6-in-order method (although oD&D allowed you adjust your stats in limited fashion after rolling them, increasing your prime requisite by one point for every two you deducted from your Str, Dex, Int, or Wis). 2nd edition included a number of other methods (4d6k3, roll 3d6 twelve times and assign the best six, points-plus-dice where every score started at 8 and you assigned seven d6 rolls to your stats, etc.).
But the AD&D rulebooks, unlike the oD&D books, have tended to philosophize about what ability scores and their various generation methods are supposed to represent. (Whether you buy into this explanation or not is irrelevant; let's just take it at face value for now, for the sake of argument.) The 2e DMG had a little section that explained, "There are two competing theories about what it means to be a PC. Are they the same as everybody else but just a little more brave or foolhardy? Or are they a cut above the norm?" The default method, 3d6 in order, presumed that PCs were just normal folks in a crazy profession. Using any other method that weighted the stat rolls to produce better-than-average characters meant that the DM ascribed to the alternate viewpoint, the PCs are special.
AD&D 1e and 3e both use the 4d6k3 method, because they follow the philosophy that normal people don't become adventurers, or if they did, they wouldn't survive anyway. The weighted die rolls represent a kind of background Darwinism that says, "we'll just skip over all the characters with bad stats who didn't survive that you would've wasted your time playing, and get right to the competent heroes." But this point of view is, well... it's bizarre to me, precisely because I came into the game with 2nd edition. 2e made no bones about its disdain for high scores and above-average PCs. Characters with all high scores were a symptom of power-gaming, of munchkinism, something to be discouraged to the extent that it made the game less challenging or inhibited role-playing.
So what we really have here are two competing viewpoints about what's "fun": having a character with high scores who feels "special"? Or having a character who starts out ordinary and becomes special through experience? As I've pondered this over the years, I've waffled between the two positions. Many, many players out there have sat at my table only to gripe and moan that having average stats, having only one or two bonuses, or having (*gasp*!) a penalty just isn't "fun". But I've come to the conclusion that this isn't a healthy attitude: it's not only annoying as crap, it also bespeaks a player who just wants to "win" at everything, a player who can't handle even the minor *possibility* of failure. Such players are often, in my experience, toxic at the game table. The advice from the 2e DMG that said to discourage this sort of behavior has proven, after a decade and more of gaming experience and hindsight, to be spot-on.
(Note that all of this is predicated on the idea that having above average ability scores, as in all 12.5s instead of all 10.5s or whatever, actually matters to characters' competence and survival. Some folks don't think that it matters much; I do, I can't be naive about that. But even still, the actual impact is less than the players' perception of it, and what I'm talking about here are the players' perceptions and attitudes.)
So my preference is for the player characters to start with average ability scores, no better than any NPC off the street. If I'm going to start a campaign at 1st level, it's really truly important that the players feel as if their characters are at "0 XP" -- they're apprentices, greenhorns, newbies, they have *no experience* as adventurers. Because, honestly, they're not even adventurers yet, never mind heroes, not until they've actually done something. Power and competence have to be earned.
Now, I'm perfectly okay with a few minor advantages, like a maximum starting hit die (because, really, who doesn't use that rule?) or maybe even the occasional fudged die-roll to simulate a bit of plot armor (as long as it's only every once in a while, and the players never find out). Max HP at Lv1 is pretty important, because otherwise the PCs will indeed think themselves too fragile and not take risks enough. Fudging, meanwhile, is always the DM's prerogative, though it takes some skill to know when it's okay to fudge vs. when fudging away a failure would actually be unfair to the players (viz. when it infringes on a meaningful choice).
But starting off the players as stronger, faster, tougher, smarter than all the commoners in the land? Nah; that's not the genre of fantasy I like to play in. Give me the zero-to-hero, farmboy with a destiny over Sir Smiley of the Shiny Teeth and Above Average Ability Scores.

It's worth remembering, the PCs are already "special" because they're being run by the players (and because they have a character class and can level up). That should be enough!