Very intereting question. I am afraid neodarwinism have no coherent answer to it - there would be probably many untestable stories of evolution of it. I don't know when the fur evolves during embryonic development of apes. I would bet on late stages. In that case I guess the neoteny would be probably the answer of my taste. Human morphological development is so to say interrupted and people look most embryo-like comparing other primates. People are also so to say born prematurely, they should have been in uterus for one year more. (and there is no physiological explanation why they are not. Those mothers who bear children longer should have had some survival advantage) So the first year of their life they are totally dependent on their mothers and we can speak about social-uterus development. It coincides with following development of language, stance etc which cannot be reduced and compared to other primates. Human are special and on my opinion their evolution cannot be explained by evolution of primates.
The problem with a bigger brain is that the head also needs to be big enough to accommodate it. Let a fetus mature for a year in the womb and then try to fit that huge head through the birth canal. It simply would not fit. Sure, we could to a C-section, but how long has that been an option? (hint: "C" stands for "Caesar"). Certainly not back when we were evolving a bigger brain.
In case you still don't realize what the survival advantage of giving birth to a child with a fully grown brain, the answer is zero. Those women would die in childbirth, leaving the baby, should it through some incredible miracle survive the ordeal, without a mother to care for it.
Other opinion is this: it isn't given fact that the head of child has to accomodate to the pelvis channel of mother. Because the other possibility shouldn't be excluded - it could be the pelvis channel that should accomodate during evolution to the child head. Of course there is correlation between the size of the pelvis channel and the size of the head of child. But the question is if other correspondences can be really excluded. We should take into consideration that by many "nest-runaway" species there is reversible change of pelvis-form in the time of birth.
Gorilla having weight 100kg deliver of offsprings which have weight 1,8 kg with brain weight 0,2 kg. Woman (cca 50-75kg) give birth to children having weight 3,2 kg with brain cca 0,37 kg. The proportion between legs, arms and trunk are by primates' youngs very similar to those of adults. By human children the relations are distorted and correspond more to embryo. It will last one year in extra-uterus development until it will be similar to adults.