Merely verifying that he was born on us soil isn’t all we want to know.
They would LOVE to get away with merely satisfying that one question, and in fact have already attempted to do so. It isn’t really a lie, and if they weren’t asked if if his name was Barry Hussain Obama on the records, he’d get away with misrepresenting himself to the electorate.
I think this is less of an issue. Lots of people go by various nicknames at different times in their lives.