My son was born in Nizhny Tagil, Russia, but his birth certificate says he was born in Yekaterinburg, Russia. It was amended by the court during our adoption proceedings to make it easier for us to obtain a copy of it if needed, as the certificates are filed in the jurisdiction of birth, and Nizhny Tagil doesn't have an international airport.
And what's more, the documents effecting the change are sealed, and any government official faces criminal sanctions if they disclose anything about it.
Well, that's Russia. There may be similar legislation in some states, but not most, I think. But the formulation I gave would not actually release any details of the birth certificate---just verify information in it that has already (supposedly) been released, but do so in a manner which actually would satisfy LEGAL requirements for proof (notarized paper statement or sworn verbal). The actual document wouldn't necessarily appear.
And frankly, I think the laws for election (especially for President) should be changed so that any and ALL records of any candidate become public. The notion of "this record is sealed", "that record is sealed", and so on is simply BS. If you're gonna run for president EVERYTHING about you should be available to the public. If you want to keep secrets, don't run for office.