Why not? Everybody would have to have their birth certificate for this, just like for a job application or security check. It's no big deal, especially if it will help send home people who are in the US illegally.
I'm not sure I understand what you meant to say here. I have never been asked for my birth certificate whan interviewing for a job, not only that, but it would create a problem for naturalized American citizens whose birth certificate is from a different nation.
There's also the fact that birth certificates offer nothing in the way of proof as to whether the individual presenting it as proof it is actually the person in the document. They are also very easy to counterfit.