That's interesting. Obviously we wouldn't need that anymore, since we have genetic testing available to us.
Although, there have been some cases where mothers have almost lost their children because genetic testing showed the children were not theirs, but more likely to be a sister's kid or someone else related to them. After months of battling, they proved that these mothers were chimeric, meaning they had at least two different sets of DNA in their body. One woman had to have the social worker inside her delivery room with her to prove that she delivered her child, and then the child was tested, and luckily showed up as having the different DNA, so that she could prove the claim that yes there was something different about her genes going to her children to get back her other children. Even genetics aren't foolproof, despite popular belief. I actually started looking into it after watching an episode of CSI and found it was true.
http://www.nytimes.com/2013/09/17/science/dna-double-take.html?_r=0
Birth certificates, just like marriage licenses, help to simplify establishing legal kinships for us. Sure there are some problems, such as people using them to deny access to that legal kinship for certain groups, but hopefully we can fix that by opening up marriage without any regard to sex/gender and even possibly finding a way to allow at least a recognition of multiple spouses as kin, even if some changes or legal measures need to be made/taken to allow for this.