1) The key to 1e feel is the meeting of the mundane with the preternatural. That is, the game must be rooted in the ordinary and even the gritty, so that there is a recognizable quality of normality in the world. The starting point of the campaign must be somewhere the players can imagine in real history or at least some approximation there of. But at the same time, this must be a world where you move from that mundane reality to a world of high enchantment and mystery combined with not a small frisson of horror. You have the overworld with its gritty reality and an underworld (of some sort) where things are strange and dangerous. Very much, the setting supports a classic "Heroes Journey" Cambellian monomyth, in which the Haven is represents one world and the Dungeon the other. The ideal experience is for the player to feel like as they are playing the game, they've stepped through the wardrobe into Narnia or crossed that realm of sleep into the Far Country.
Modern D&D for all its merits just doesn't do it. "Honor Among Thieves" is not 1e feel, as the whole world is high magic and filled with wonder and right from the beginning the protagonists are partakers in that high magic world. Modern D&D from 3.5 on (played RAW or according to the apparent intentions of the designers) isn't fantasy at all but Swords & Capes Superhero genre. And while you can get there in 1e, you only get there in 1e by completing your Heroes Journey. Becoming a superhero in 1e D&D requires you passing through the Road of Trials.
2) First Edition feel is process simulation over game most of the time. If you look at Medusa, you turn to stone no matter how bad that works as a game mechanic, because in the genre and the stories you are trying to simulate that's how it worked. The first edition game masters and player base were "Playing at the World" as Jon Peterson so elegantly put it, and you can see this in a lot of the Dragon articles of the day or in the sort of stuff that Gygax felt the need to cover in the 1e DMG - everything from tax collection to siege mechanics to how rapidly the different races could mine different kinds of stone.
3) First edition feel is player skill tested ahead of character skill. Failure is the expected state and the goal is to try to transcend that expected result and "win". The players of 1e AD&D were coming directly out of a competitive wargaming background and initially the game was considered to be an imaginative extension of that. This is why "Tomb of Horrors" is such an iconic 1e feel module. Your choices matter far more than the playing piece you are playing. That isn't to say the character whose role you are playing doesn't matter, but ultimately it's secondary to the idea that careful and thoughtful and creative play should carry the day. The character in a sense is only there to protect you from your own bad decisions. The character doesn't make decisions for you, and does not innately have the ability to solve problems just to resist failures. This does get lost at times and you do see grindy luck based design in a lot of cases, but it never really gets completely lost the way that it does in later editions. But it's important to note that however much the players and GMs enjoyed the idea of superior play and skillful play, it was secondary to the simulated world. If the world was too false and artificial feeling, then success in that world didn't matter anyway. The goal was to feel like you were succeeding in the world of fantasy and story, and not to feel you were winning a board game.
