The great intelligence fought the doctor. It gathered thousands of the best of human minds - healthy and functional, for many years. It traversed the universe and was able by guile and cunning to bring the doctor to Trenzalore, and to enter his time-stream. It was the "magnum opus" of defeat of the doctor at every point in his time-lord timeline. In a way it was meant to convert the entirety of space-time into darkness.
But the great intelligence was a survivor. It was regularly about living to fight another day. It did not have problems casting off chaff to achieve the long-term goal. Victory and not suicide was the long-term goal.
And also, Clara entered the time-stream after the intelligence had been broken into echoes and fragments. River said what happens when one enters the time-stream.
It is likely that the super-human "great intelligence" had as its opus not only the defeat of the doctor, but its continued post-doctor eminence in the transformed dark universe.
And also, Clara entered the time-stream and displaced the great intelligence.
If the great intelligence had an exit strategy for the time-stream, one that increased its capabilities, then Clara to some degree fell into that trajectory, and displaced it. She was not only in the asylum of the daleks, but imbued with an intelligence that transcended the intelligence of both the doctor and the daleks. Why would a fraction of her have more than the sum of her? Mathematics doesn't work that way. When you cut a pie in multiple pieces (pie-ces) then each part is less than the whole. When you take apart a watch, each gear and spring has less function than the whole, not more. When you take a finite human apart - why would there be parts with more function than the whole?
What if the smart one was the great intelligence? It was, wasn't it - smart? Did it pave the way - only to be bumped out of the roller-coaster at the last second by Clara?
So is Clara, especially rescued from the time-stream by the time-lord, still "merely" human? Is she super-human? Did the great intelligence build "cheats" to make itself supremely powerful, only to have Clara displace it from its throne, and unknowingly take on some part of those attributes herself?