No. There is no conflict between the 14th Amendment and Article II in this discussion. If Obama was indeed born in the US, that would be the end of the argument; he'd be a "natural-born citizen" and would be constitutionally qualified. Problem is that his place of birth has not been documented by a bona fide birth certificate or officially certified copy of such true birth certificate.
True, but there is no one of relevance right now who can or is willing to force this issue.