Puzzle gameplay is about pattern recognition and planning, third person action/shooter is about fast reactions, multitasking and target prioritization, they are essentially at odds with each other, can they satisfactorily be combined? I want this game to be good so much, but I'm I'm not so sure that they can pull this off.

I think if you could tell her to do something separatedly from the player it could lead to more more dynamic gameplay, kind like a Bioshock protect the little sister moment, also it would improve the feeling of she being her own character instead of just an extension of the player character.
The minigame could still be used to speed up whatever she's doing while you juggle the monsters, but maybe it wouldn't necessarily be a hard requirement for even being able to fight at all, the way it is not, the 2 main gameplay mechanics are not connected in parallel way, the puzzle is just a gate to get through to the action gameplay.