Actually, there are statutes in some states allowing just such verification. But not a single state official bothered to do so. Not one.
Not quite so - but this is amusing ...
Roger Calero ran for President as a Socialist Worker's Party candidate in 2008 - although he was born in Nicaragua in 1969. He actually made it on to the ballot in 5 states - and he was on the ballot in 2 others, but the Secretaries of States for those states found out about it and removed him.