If we define OSR as a style of gameplay that focuses on dungeon crawling, resource tracking, combat as war, killing monsters and taking their stuff, 2e does that fantastically. Because the mechanics of 2e *are* the mechanics of 1e, it is literally built for that style of game.
2e has a lot more rules though. That's because AD&D started to position itself as a major roleplaying campaign system starting with 1e. In truth the seeds of it were there in OD&D, it was just never fleshed out. AD&D 1e really expands the fantasy medieval campaigning, but it is frustratingly incomplete. Part of it is due to Gygax's haphazard writing, part of it is the practical constraints of how much content can actually fit in the book. 2e takes that and builds on it, but instead of concentrating on fantasy medieval wargames campaigns, it turns into a sandbox world simulator.
Most of AD&D 1e was written in the editors room between Gygax and Kask, and it shows. Many ideas that seem good on paper turn unworkable on the table (re: polearms). By 1989, AD&D had been out in the wild for 10~12 years, and the staff at TSR had a long time to play and understand its shortcomings. It was with the benefit of this experience that Zeb Cook revised and modified the rules of AD&D for 2nd edition. Though the suits at TSR get a lot of hate, one thing they did right was stress that 2e remain completely compatible with 1e modules and splat, which reigned in any of the more extreme changes that Zeb Cook might have made. I feel that if Gary Gygax had released his version of 2e, it would be something very different from what we now call OSR.
By 1995, the bottom was falling out from under TSR. Splatbook bloat and competing product lines, inventory mismanagement and vanity investments by Lorraine Williams to fill her own coffers had caused TSR to slash most of its product lines out of panic. Ironically, this led to some of their best products in decades. The "Introduction to Advanced D&D" set is a fantastic intro not just to AD&D but the whole roleplaying hobby. And it focuses on town and dungeon crawls just like those OSR purists claim "real" D&D is like. If you need proof that 2e is a fantastic game for OSR style play, just read and play that set. It includes a subset of the full 2e rules, so you don't have to go through the whole core books.
I'm not saying 2e is perfect though. It still contains a lot of jank and clunky mechanics. like any RPG, it needs to be tailored and customized to the table in which it is played. And it has that splatbook problem: some are great, some are terrible, some in between. When I run 2e, my focus is on using it to create "realistic"/believable worlds to play in, and populating it with adventures and danger. (side note: I think my danger quotient might be too high) It is unlike 5e, which fundamentally fights against the OSR style of play by reducing challenge or outright removing it, ignoring resource tracking and focusing on "balanced" narrative encounters.
But if we define OSR as "a secret club that exists in the minds of grognards and wannabe LARPers that begins and ends at arbitrary points in time", then 2e is not and cannot ever be OSR, nor do I see why that should be anything to aspire to.
On a personal note; I do not care whether 2e is considered OSR or not. I enjoy the game because it does what I want out of fantasy adventure gaming. It is clearly old school, compatible with Original D&D (the white box) and AD&D 1st edition, and not part of the 3e and above line of game design.
