hmmm, I think it depends , my mother has alwasy been cold and distant only when I grew up and she didnt have to take care of me we became closer , my dad on the other hand loves children , he always wanted to go camping , chat and do all those super cliche father son activities wich I HATE! my mother wanted me to shut up because she was watching tv or talking on the phone.
I believe the bonds are just diferent , perhaps the mother is more obvious , but men change when they become fathers , when they see their child for the first time , etc.
and about the social construct yes, fatherhood could be a social c. but women being super loving mothers is a social construct too , even since they are little they are taught their ultimate goal is to reproduce and not every woman wants that. men on the other hands are taught getting her pregnant is the worst possible consequence XD haha and it isnt what all men think.
if there wasnt a natural father instict , no men would want to have children, EVER.
and I mean ... some species eat their children , and some human mothers kill their babies too, its only natural , haha.
anyway , this exemplifies my answer about oposites and partners at the same time. and how both are necesary.