Even accepting that, it doesn't necessarily follow that enforcement lies at the state level. As I noted (Post #39, above) a Presidential candidate either meets the Constitutional qualifications or not. So a candidate can't be ineligible in Alabama, yet eligible in other states. But positing that enforcement of the Constitutional provision vests in the respective State secretaries invites inconsistent results and chaos. (And there is the obvious potential political mischief that could ensue were a Secretary in a key "swing state" or delegate-rich primary state to act to keep a particular candidate from the opposing party off that state's ballot).
While Article II is silent as to enforcement, guidance can be gained by viewing the Constitution as a whole. Relevant in that inquiry are the Supremacy Clause, Congress's impeachment powers, and the powers and duties delegated to the Congress in the Twelfth and Twentieth Amendments. The better argument then is that any Article II enforcement lies with the Congress (post-election) or perhaps with We the People (pre-election) through refusal to vote for an ineligible candidate.
I would also assert that it is axiomatic that the enforcement powers of constitutional provisions automatically trump any state laws which attempt to interfere with enforcement, i.e. Record privacy laws.
Once you bring state records into consideration, the Full Faith & Credit clause also comes into play. It is axiomatic that the state which has issued a record has the ultimate say on the content and validity of that record. Hawaii had made repeated affirmations that its vital records indicate BHO II was born in Hawaii. The FF&C clause makes it hard to sustain the argument that the question should be re-considered ("enforced") upwards of 49 times at the state level.
This ruling from 2009 went to the Supreme Court of the United States which denied Cert., so it stands:
Barnett, et. al. v. Obama, et. al., U.S. District Court Judge David O. Carter: “There may very well be a legitimate role for the judiciary to interpret whether the natural born citizen requirement has been satisfied in the case of a presidential candidate who has not already won the election and taken office. However, on the day that President Obama took the presidential oath and was sworn in, he became President of the United States.
Any removal of him from the presidency must be accomplished through the Constitution’s mechanisms for the removal of a President, either through impeachment or the succession process set forth in the Twenty-Fifth Amendment.
Plaintiffs attempt to subvert this grant of power to Congress by convincing the Court that it should disregard the constitutional procedures in place for the removal of a sitting president. The process for removal of a sitting president—removal for any reason—is within the province of Congress, not the courts.”—U.S. District Court for the Central District of California, October 29, 2009
http://ia600204.us.archive.org/1/items/gov.uscourts.cacd.435591/gov.uscourts.cacd.435591.89.0.pdf