If you make it to be that big of a deal, no woman who loves you or cares for you would object to you tagging your child with the patrilineal surname you have. However, most women who don't take this as the norm or some trivial matter, and attach value on the possibility of their chilf having their names, would be quite sad or heartbroken inside.

I see this thing from two perspectives, one of a daughter, another of a future mother. So what my mom gave birth to me and my father didn't? That doesn't make the strangers know who my mother is. But I want them to know that, from my name. And this societal norm of tagging me with my father's patrilineal name feels forceful, that's not even his own name. I would rather like one of the two following system as default: women will have their own matrilineal surname, daughters will get their mother's surname. Or, the second default can be that kids will have both parents' (or, one in case of single parent, or, more than two, in case of more than two parents, which I don't think is legally a thing in most places, still) first names after their given names. Or, a third default will do too. No surname at all. All these systems remove the sexism bias, as well as the casteism and class difference bias.

Second perspective is that of a future mother. I would very much like my kid to be known to people by my name too, just like they will be known from their father's name. Once I pointed this out to one of my male colleague and he told me this is a trivial matter, world is unfair to men in various ways, so women should not make a fuss about such a trivial matter and let men have this their way. Well, women are also not without problems, and many women want their children to have their names as a default, it's very selfish, narcissistic and entitled of the men (who think like this) to tell women to make way for their feelings and sentiments at the cost of our own. And if it's not that much of a big deal really, then why these men making a fuss about this in the first place? Why not let the women have it in their own way? So, it's not a big deal for women but is for men? Why are these men, not living in a woman's shoes, deciding for us that it's trivial for us?

Some other "arguments" that I hear about this issue are as the following:
1. Children get their inheritance mostly from the fathers. So, if you want money from papa, you tag yourself with his surname. Wonderfully narrow-minded argument. Do they mean fathethood is conditional? If you don't take father's surname, you don't get his care, protection, money etc.? I know fatherhood is a social construct to pass down inheritance patrilineally, but fathers nowadays are also considered as equally loving parents just like mothers, so much so that we are legally allowing entire child custody to men in many cases, legally accepting men to be completely capable of being a homemaker and primary care-giver, entitled to receive alimony and maintenance from ex-spouse etc etc etc. (which BTW are good things, and should be like this). Then why are we still arguing with such weird biased mindset that if you want dad's money, take his surname? I can form a similarly weird argument: Most women around the world are the primary homemakers and careguvers, so if you want mom's protection, her care, take her surname. Sounds like motherhood is conditional, right. And should not be the case. That's why both these arguments are ridiculous.

Second weird thing that I hear as an "argukent" is that if the kids don't have father's surname, then there will be no way to know if a man whores around, and fathers babies from different women. Then the "illegal" children won't get his money. Hence to prevent that, all children should have dad's surname. This argument doesn't even deserve a rebuttal. If a man is so rotten, it's better to not take any pf his money, he can go to hell with his money and his penis.

Third "argument" that I hear (and this is argued by many women too, not that no women provide or agree with the above bs too, but this third one is more common from them) is many kids actually like their father's surname. They are not "so sensitive" about gender equality or unbiasedness. So, the current system works for them. Well, the obvious thing that they miss is that the default should always be unbiased. Now, everyone should always have the choice to change their reality in their own way. But what happens now is the opposite. When a child is born and their name is decided, the government automatically assigns their dad's surname to them, society also think that is the practical thing to do. Hence, it's biased now, which should not be the case.