Given that only women gestate and lactate, however, and that nature has evolved a hormonal cascade to make women want to care for their children, I wonder if it will always be the case that more women than men want to play the role of primary carer, simply because biologically it will be the woman performing that role for the first 9 months (with baby in utero) before the man can even get a look in.
Obviously there will always be exceptions and a society that encourages men to take equal responsibility will do much to equalise numbers, but I can't help feeling that women will always outnumber men.
I don't actually think that's sexist. I think it's acknowledging that men and women are different biologically, which they are. What's important is that the difference does not mean inequal. I think, for example, that rearing even very young babies is about much more than simply breast-feeding, which is why there is no reason men cannot make just as good primary carers and society should be restructured to allow that for those parents who want it.
However, I do feel that it is right that women should have a more significant status for a short, finite period to reflect the fact that their involvement in the creation and production of that child is far more involved than the father's. In some cases the woman's body is irrevocably changed for the worse, not to mention her mental health in some cases. Women take a much, much bigger risk having children than men can ever do.
As the child grows, this 'superior' status changes and equalises, but immediately post birth, I think women should retain the greater 'right'. In cases where both parents want to be involved, this shouldn't matter and society should allow both to be as involved as they want. However, in cases where the child is born and the parents separate or were never together in the first place, if the mother wants to be primary carer and it's not obvious that the father would be a better choice (because the child's needs has to trump both parents'), I feel the mother should have the greater claim.