Hmm, that's an interesting idea merrymouse, and of course you are right about genetics as such being modern.
But I think the idea of the biological connection to parents is not new, and that comes down to the same thing. The legal responsibility of the parents is related to that connection in whatever material terms you understand it. It's not just legal.
I would argue that this was considered true for fathers as well for the most part, the problem was identifying them as such. The father could do a runner as you say or the mother could identify him and he could refute that and it was difficult to prove otherwise - these were attempts to prove that there was a parental responsibility or refute that such support was due. That was a major advantage of marriage, it was a sort of guarantee that the father would be counted as a parent with parental obligations.
One paternity testing became a possibility it began to be used to establish the parenthood of the father and the legal obligations that came from that, and I don't think that would have been the case had the idea that biological fatherhood carried obligations had not already existed.
To me it looks like all these attempts to change the birth certificate toward being a document that reflects social arrangements or who legally "owns" the child are about unhitching it from the biological basis of the parental relationship. It's not about the limits of what we know or even accommodating unforeseen circumstances like the father was really someone else. Once it becomes accepted as normal that the birth certificate isn't about who are the people who created you, it's easy to say, why treat the mother any differently?