At about the same time I was looking at Tadhana (maybe even a little earlier), I was also reading through Macchiato Monsters by Eric Nieudan. It is much more faithful to the OSR ideal than any of the other games I have examined to date. According to the author, Macchiato Monsters started out as "an adaptation of The Black Hack (Black) with the classes and magic system from Whitehack (Mehrstem), both inspired by the original fantasy game (Gygax and Arneson)" but later became its own thing. He also cites Into the Odd and Maze Rats as inspirations, which becomes important later on.
Macchiato Monsters is very streamlined compared to 5e, but it is similar enough to at least look familiar to 5e players. Characters are built around the standard six stats, but there are no stat bonuses. Instead, all stat tests are d20 roll-unders. Traits are used to encapsulate concepts like species, (sub)class, and/or background and grant advantage (lowest of two d20s) on stat tests where the character's perceived talents would be useful. During character generation, players choose types of training (martial, specialist, or magic) to provide them with the tools necessary to play the character a certain way. Very simple and super-functional, in my opinion. The use of "risk dice" to manage resources and to keep things moving (kind of like The Angry GM's Tension Pool) is also really nice.
The combat system is designed to be quick and deadly, with only one die roll per player per round. The player rolls to hit, and if they succeed, they inflict damage for all of their available attacks. If they fail, then their opponent inflicts damage for their attacks instead. The GM rolls few or no dice, farming that task out to the player. I personally prefer not to roll dice as a GM, so this was a big plus for me. Things can get somewhat confusing when there are multiple characters ganging up on an individual, confusing enough for the author to later post an explanation on reddit.
After starting to map out a setting to use with these rules, I eventually abandoned the project for a pretty silly reason: I don't like d20s. Dice mother-flipping hate me, and over the years, the feeling has become mutual. Seriously, my results with d20s are always statistically improbable, which is why I've spent so much time creating and researching card-driven systems. Before that, I gravitated towards games with bell curves where rolling multiple dice causes results to almost always be in a realistic middle range. The problem with bell curves, of course, is that play gets kind of dull when one can predict everything that is going to happen with at least 68% accuracy.