I don't understand your logic on this one. Are you saying that men will not care for a child that doesn't have his name? Do you really think men are so vain and weak? Because that is patently false on variety of levels. Not only is giving a child the father's name a relatively recent thing (less than 10k years old), in many, many countries today this isn't necessarily the default. In modern matrilineal cultures children take the mother's name, and fathers still manage to love their children and care for them. In certain South American cultures where partible paternity is common a woman who wants to get pregnant has two (or sometimes more) men "father" her child. All the men involved care for the resulting offspring which often leads to greater life expectancy.
In addition, a lot of places around the world it is actually illegal for a woman to change her last name so children often bear the names of both parents or as in Greece, the parents decide if the child will have the mother's name, the father's name, or both.
I didn’t change my last name when we married. My son’s middle name is my last name and his last name is my husband’s. I had no issue with that, and I think in general, people should do what they want to around this, but it’s an American conceit that this is somehow timeless and universal.