FATE is plenty mechanical. It has the usual bonuses and skills and stuff. It's just that it lets the players apply a lot of extra bonuses on top based off their character traits, and the game runs off this back and forth.
You know what? Have you ever played D&D 5e? You know how characters have personality traits like drawbacks, flaws, bonds, ideals, etc? And that the game recommends letting you get inspiration or roll with advantage when you lean into those? It doesn't happen often because it's sort of an afterthought in 5e, but that concept came from FATE. FATE builds the entire game around it. One of the "problems" with knowing when to give inspiration is what counts as leaning into it. If a player is very mechanical minded they might want to know EXACTLY how many feet they need to rush towards someone for it to count as a reckless charge, if their character is supposed to be a hot head as their flaw. In 5e most of the time it doesn't matter, but FATE has your main character trait based on concepts like this.
You pick an aspect, which is basically a description of your character, a title. Let's take my avatar, Perfect Cell. If I was using him as a character, I might make his aspect "the perfect being", that being a genetically engineered super fighter who has traits from all the other martial arts characters. There is no list of aspects, it's just whatever you want it to be and the GM approves. OK, so what does this do exactly? Nothing, on its own.
But when it's time to do some sort of action, in the same way a player might try to pull from his backstory to ask if it helps, like they might say "hey in my backstory I used to be raised by orcs, so would that help me understand these orc bandits we just ran into better", I might say, "as the perfect being, I have the same martial arts knowledge as all these other fighters, so it helps me with this roll to learn what martial arts style this other guy is using." The difference is in the former, it's purely the player basically asking if it's OK and the DM allowing it, while in the latter the game encourages the player to just lead with it and only if it's really crazy would the DM disallow it.
The DM can also draw on your aspect to put you in tight spots, using the same aspect -- a good aspect should have a good and bad side, not just be all good. For example, as "the perfect being" who has all of the other characters' traits, like Saiyans, the DM might say that I have to spare the characters I just beat because I would want to have a better fight. And I would have to go along with this (or pay some extra resources to overcome it, sometimes).
So as you can see, it requires the player being a good sport in not trying to constantly push the things they get out of their aspects to absurd lengths while at the same time using them to get most of their advantages, while also being the source of their disadvantages. And this is a negotiation done on the player to GM level rather than something the character has any idea about or decision about, in-game.
It is basically mechanizing how your character concept would impact the game without breaking it down to a ton of different separate skills and abilities you build out.