A question for the Patriachal Social Model

I just have this burning question- "Why are family names only passed down through sons?"

This is applicable to almost every society in the world. Well, technically, you can never doubt the mother of a child, but you can certainly doubt who the father is... So if the child's genes are in doubt, why should surnames be passed down from the father's side instead of the mother's side of the family?

Anthropologist would typically say that it has to do with the social organisation-Pastoral and Agrarian societies tend to be paternalistic because males do the hard work but Horticultural societies tend to be maternal.

That bullshit aside, surnames are meant to identify the offspring as members from a certain family. Only the women can assure that their children carry half of the family genes, not the men. ^^ So is it time to revise that antiquated system and redraw the family tree based on maternal links?