I believe that there should be one rule for everyone within a society. That way you can trace back family lines and so on. It would make a lot more sense if the woman always gave her name to the children (and husband), since she has traditionally been the one to stay at home and raise the children (whereas men could in theory come and go). It was also self-evident that she was the mother, whereas even today the man would need a DNA test.
In modern societies (with divorces and single mothers) we also have the paradox of a woman having two children from two different fathers, neither of whom will share her own name (if she decided not to marry either of them or just kept her maiden name), whereas a divorced man can remarry and have children with the same name that have never even met each other.
There are some societies where property, name and lineage are passed on from mother to daughter, and men are seen as appendices to those families. It makes a lot more sense as a system.
But to answer your actual question, no, I would not accept that if I was the only one doing it. But I would be willing to do it if everyone else did it too, or vote for some sort of change in legislation if it ever came to that.
There is a woman I know called Kerry Grist. She has two children, one is called Ben Needham (you may have heard of him, he went missing 18 years ago) and the other Leighanna Ward. Both children have the same father (Simon Ward), but since she was not married to him when Ben was born, he got her maiden name (Needham). Leighanna was born after she got married, so she got her dad’s name (Ward). And Kerry remarried recently, meaning she now has her husband’s name (Grist). So one mother, two kids, three different surnames. And they don’t even have a different father.