The vast, vast majority of awakenings come from institutionalized teaching.
We can, in fact, examine our decidedly-abnormal party here and still find similar patterns;
Illiyeen because she is disgustingly self-reliant managed to essentially put herself through a basic spy training course through careful, measured observation of patterns of behaviour, deliberately expanding her horizons and teaching herself French so she could listen in people's conversations, and months of watching, waiting and learning all in pursuit of the truth.
Henriette possibly Awakened due to 'my parents are dead' trauma (a so-called Batman Awakening), but recent evidence suggests quite strongly that it's also linked to the whole Twin Souls thing and the fact that at the same time the Computer was carrying out hypertech Incarna unethical experimentation on Henrietta trying to force an Awakening.
Kessler followed an entirely typical Iterator steelhead path to Enlightenment - joins as a mook, probably learns some linear sorcery while studying for his degree in asskicking, Awakens as an enlightened scientist after years of training.
Elsa followed an entirely typical Virtual Adept path to Awakening - joins as an outcast (Russia not exactly being entirely tolerant of her sexuality) who hangs around the cyberpunks, trains as a consor and linear sorcerer hacker, Awakens after years of training.
Donald followed an entirely typical Ecstatic path to Awakening - falls in with a certain crowd, sex drugs and rock and roll, gates of perception get blown wide open after a few months. Notably in the Control timeline, he follows an entirely typical Iterator scientist path to Awakening.
Rose... uh, who knows? We don't really know how she Awakened. It was either from being a mental newborn exposed to Progenitor primers and thus accepting them fully, from mighty sorceries worked on her when they were trying to make her into Reina Lior, or from all the trauma inflicted on her over her life.
Serafina, amusingly enough, despite being the only party member who's gone through the ludicrously expensive Damien education did
not Awaken thanks to the countless Primers and teaching she was exposed to, but did it during the school holidays because she was a desperately lonely little girl who really wanted a friend and to get to do things with her parents.
...
So of the party, we have three institutionally taught, one self-taught prodigy, one emotional-distress-primed-by-teaching, one unknown, and one emotional-distress-or-possibly-bleedthrough-from-inhuman-godmind-forcing-her-twin-soul-to-Awaken.
Yes, that does mean that it's possible that just like all good superheroes, the Computer accidentally created its greatest nemesis.
(Meanwhile, hmm, Jane Clarent is Technocracy-taught, Antoinette is Damien educated, the EXEMPLARs are mostly Technocracy-taught apart from Maria who went AWOL and had a consciousness-expanding experience in a world where she wasn't constantly told what to do and also started a war, and Elissa/Alice was... uh, a psychic at ground zero of her crazy mother botching an Enlightenment 6 Seeking. And was already primed by Technocracy teaching. Hannah had a pretty typical Orphan Awakening finding weird things on the net and being led down the rabbit hole by scattered rumours and tales, while Melody was visited by a magical fairy who told her that she was a magical girl, possibly caused by her freakishly managing to track down a VHS of the lost Episode 0 of an early 90s magical girl show which never got an English release, managing to find something which could actually play VHSes [1], and watching it in full [2])
(Oh, and Aqidah became an Enlightened Scientist because she's really, really good at computer games [3]. So unfair.)
[1] Which was only marginally easier than finding the VHS in the first place.
[2] Melody has taught herself Japanese so she can watch raws. Many people are amazed that she could do something like that. Many more people just wish she could put that effort, focus and capacity to concentrate into actual academia.
[3] Intelligence 6, Tactics 4 (Strategy Games) good. UFO: Extraterrestrial Combat, Long War difficulty, Iron Man mode, All Mutators, 4 casualties good.