This is the strangest thing to me: using "first edition feel" as a selling point.
Read the 1e AD&D PH and DMG. Then read any other edition. Every other edition will be better organized and better edited and arguably better laid out. Every other edition will be more professional. But the 1e AD&D will have the highest grade reading level, the most archaic language, and the most entertaining writing. Arguably, even the basic books are written in a more elevated style than modern editions. The feel of the 1e books was as if they were ancient lore filled tomes. Gygax never assumed that his audience was aught else than largely college educated history buffs and war nerds much like himself. The modern books have the feel of rule books, and have the disadvantage of knowing that much of their audience is 12.
Probably the single thing for me that other editions have never quite captured was Gygaxian naturalism. Gygax described and was seeking to describe a vivid world that could be generated procedurally. There are reasons why the old monster manuals with their percent in lair, number appearing, and treasure tables still fascinate, and why the procedural dungeons in the appendix of the DMG still amaze. The older books read like guidebooks to some world beyond a magical portal. The newer books read like game books.
And there are reasons why the old modules still make the lists of best modules ever some 30 years later, and those reasons are not just nostalgia or what came first. I'd argue that early TSR existed in a brief window of time where pen and paper role playing game could attract the best talent in game design, game writing, level design, and art because the tools to develop sophisticated games had been invented but computer games hadn't yet matured to the point that they dominated the market for talent. It's very hard for any gaming company now to hold on to the sort of talent you'd need to out do early TSR because there is basically no money in PnP RPG's and computer games are like a trillion dollar industry. The entire PnP RPG industry is a labor of love.
When I was younger I used to dwell on the flaws of 1e AD&D. I, like so many others, dreamed of creating a better system. And over the years so many people tried and failed. Hit points, we were told and believed, were unrealistic and stupid. Now I hear that, and I can hardly stop from rolling out of my chair laughing. It reminds me of those stories about publishers turning down Harry Potter, or the maker's of mechanical calculators turning down the electronic calculator. Hit points have been one of the most wildly successful inventions of the last 100 years. They are incredibly pervasive because they work and work well. Vancian magic we're told is 'unrealistic', as if there was some sort of realistic set of rules for magic. And so on and so forth. There have been a lot of fantasy heartbreakers out there that were going to do it better, but mostly no one ever played them. That's not an accident. It's actually a very rare situation where the first to market remains the industry leader for any length of time. Being the first to market is often a disadvantage in business, because your competitor can then respond to your moves and your lessons learned when you went into the market in the blind with nothing more than a prototype. D&D got a lot more right than it got wrong.
The adventures commonly employed inexplicably over-powered traps and monsters which didn't have a lot of reason built into them - resulting in a lot of paranoid PCs.
There are of course many examples of bad design in older adventures. But what you call reason was in its lack experienced in play as the numinous wonder of magic. Magic didn't always have to make sense. The Voyages of Sinbad, the Grimm's Fairy Tales, and various the myths and legends that were the source material of both D&D and the pulp fiction it was most directly inspired by didn't always make sense. Why is it that cursed Princes forced to take the guise of animals also as part of the curse seem to gain fantastic magical power? Doesn't seem like a very effective curse if you think about it. But the point is that it is magic, and magic of a very raw, primal, and untamed sort. So if you go into a Gygaxian dungeon and find a tree where gems are growing on it as living fruits, and you have to opportunity to eat still warm and living raw rubies and gain a random magical effect - this doesn't necessarily make any sense at all - but it is the stuff of magic.
I won't ask about rule subsystems, because I doubt that there was a general system.
The universal system has been the dream of RPG game designers since the 1970's. And you know what? At this point I think we can definitively say it's like the alchemists dream of turning lead into gold. Even if you could do it, you wouldn't want to, because the cost is more than the reward. I now believe that disparate rule subsystems are inherently superior to unified systems. The reason for that is that different things you are simulating have different salient qualities, and as such inherently are better simulated by different abstractions. You can see this when we appraise different rule systems and talk about what each rule system does better than a different rule system. That's because each simulated focus of play has different needs in abstraction because each simulated task is in fact very different in the real world. Tactical skirmishes are different than chases which are different than investigations which are different than negotiations and so on and so forth. Sure, you can find commonality between them and use a common abstraction for each of them, but in doing so you tend to be diminishing the very things that make each situation unique and likewise diminishing the very things that make the simulation believable and engaging. The correct solution is disparate subsystems. Computer games figured that out long ago. Gygax clearly figured that out organically by the evolving process of play. Yet PnP designers continue to pursue unity of mechanics like the lost Seven Cities of Gold. They might as well be flat Earthers for all that they notice what's happening.
Fantasy has become a very incestuous genera that over the years is increasingly largely drawing on itself for inspiration. Everything ends up being self-referential. D&D's tropes are now largely drawn from D&D's past tropes. Video games largely draw from other video games. It's created a sort of mythology of its own that is shared across all of nerddom - consensus fantasy. But ultimately, all of that consensus is set by 1e D&D and the world it created, and it's D&D that is both the root source of most modern fantasy and the bridge between modern fantasy and the older sources of myth - Tolkien and his inspirations, raw pulp fantasy fiction like Howard's Conan and Burrough's Tarzan and John Carter, and the vast body of ancient world myth and legend. That nearness to something ancient and resonant is part of '1e feel', especially for those of us that didn't grow up exclusively on post-D&D fantasy.