Births in the U.S. are routinely registered by the hospital with a city or county vital records office. The parent doesn't have to do much except sign a form. And if for some rare reason there was no birth certificate, a person can still prove his birthplace and U.S. citizenship with other documents -- baptismal records, hospital and school records, sworn statements from relatives and other witnesses, etc. He's not out of luck forever.
http://travel.state.gov/content/passports/english/passports/information/secondary-evidence.html
But in the example of the Indian parents in the U.S., the parents have to travel to the Indian embassy or a consulate -- only six in the entire country -- within one year of the birth to preserve the child's Indian citizenship. That could be quite a burden.
If birthright citizenship was eliminated in the U.S., it sounds like proving a child's citizenship would require the parents to take the baby's standard birth certificate
and their own citizenship documents to a federal office somewhere -- non-existent now, so a whole new agency would have to be created -- like applying for a passport. A certain -- maybe substantial -- percentage of parents would fail to do so for a variety of possible reasons, leaving the kid in a legal limbo which would have to be resolved somewhere down the road, probably by the courts. I suspect that the administrative and legal complexities and burdens would outweigh any possible benefit to ending birthright citizenship.