Fate can do this!
Many of the Fate systems I've used and studied talk about how to highlight "discover what the GM has planned" play styles. I'm going to describe the various tools and strategies I've used and learnt which bend toward your goals. I've never gone quite as far with them as you will, but they're robust and my experience with them makes me confident they'll work for your situation.
First off, Fate is really cool about leaving stuff blank to get filled in during play. It doesn't degrade from play at all! In fact, I generally encourage my players to do it because then they get to say how their character is exactly the right (or wrong!) person for the story they're in. I first learned this from Aeon Wave, but my pre-made Doctor Who character sheets for FAE are a more aggressive example of this in action.
PC creation: Instead of aspects hinting at a relationship with another PC, Behind the Walls suggests that PCs have aspects describing their appearance, past, or aspirations. Honestly the "crossing paths" phases of character creation never worked well with my group, and when we want relationship aspects we have more success with something like the "Shared" aspects in Masters of Umdaar.
Another way to define aspects is by areas of competency: atomic-robo asks me to define three broad skill modes my PC is good at like "banter," "fighting," and "sneaking." Then I come up with an aspect for each describing something specific about how he excels in that particular arena. (ARRPG's specific skills/aspects iteration of "modes" will be genericised and made available via SRD soon; I'll link to it then.)
World creation: Until Fate Core, the Fate engine didn't have a "shared world creation" default. It was always attached to an existing setting, and it continues to be used that way in many publications. As a GM, I've successfully run games where during world-creation I say, "This is a thing I need to have control over," or "I've got a cool surprise in mind but it means I'll need to ask you to make changes to the world ideas you're proposing." This works in direct proportion to how much my group trusts me to make choices that are fun for everyone.
Fate is happiest when there's big blank spaces for players to fill in during play, but exactly what kind of space that is varies from game to game. It doesn't preclude a GM guiding the narrative around a secret. Indeed, every version of Fate I've ever read has a section on how the GM can/should keep secrets.
Secrets in the game: Behind the Walls talks about many kinds of secrets and how to deal with them. It breaks them down into world secrets, NPC secrets, and PC secrets, but the basic advice is pretty universal.
The GM can use an unknown secret to give a +1 to difficulties and rolls where the secret is an obstacle or benefit as appropriate. This gives the secret mechanical weight without revealing it.
Eventually a secret must be exposed. Unwritten features a skill and scene mechanic which combine to give structure to PCs' investigations of secrets the GM has created; at its simplest Fate Core would replicate this by using Notice or a similar appropriate skill and asking 1 question for each shift on the roll, then using other skills to change the narrative so a new investigation roll can be made and new questions can be asked.
Often, however, waiting for players to do exactly the right thing to figure out a secret leads to overlong waits and sucking the drama out of the discovery. So I (and Behind the Walls) suggest something which I first found codified in the apocalypse-world Engine:
Rather than waiting for a secret to be uncovered, the GM should reveal a secret at a certain "trigger" event. Specifically, when the players' actions have brought the narrative to a point where one of the following is true:
- It increases the immediate stakes.
- It explains, rationalises, or justifies exposition.
- It will elicit a response from the table.
In conclusion: Fate doesn't handle mysteries in quite the traditional GMing style, but it's very amenable to secrets and mysteries. I suggest reading Behind the Walls, Aeon Wave, and Night Fears; they're all (free to download!) examples of Fate with a strong focus on secrets the GM knows which the players (not just the PCs) do not and are expected to be revealed through play.
And don't worry about the character creation phases. They're the first guidelines my group threw out and we've never missed 'em.
I can't consider this answer complete unless I mention that the best mysteries I've ever run were in Fate, but weren't anything like what I've described above. Instead, I set up questions and then invite the players, through their PCs' actions, to slowly declare and invent clues which eventually synthesise into a reveal nobody expected. This kind of play gives everyone (including myself!) a chance to go "oooh" and be surprised, but also gives everyone a chance to be part of orchestrating the surprise and its revelation. Atomic Robo and Unwritten have codified a mechanic to support this, which is described in the Fate SRD as Brainstorming.