Suppose the AG of Arizona was not sure of a candidate's bona fides. Some legitimate doubt existed.
He would then order the candidate off the Arizona ballot. The candidate then has the choice of contesting that action in the courts. He becomes the "Plaintiff," the AG or other election official, the "Defendant." BTW, this happens frequently in state and local elections. E.G., Rahm Emmanuel's battle to be on the Chicago ballot for mayor!
The court can
(a) support the plaintiff (the removed candidate), or
(b) The defendant, (The State AG or other Election Official.)
(1) The Defendant (in this case the State AG) is free to appeal, but in practice rarely does.
(2) If the decision went the other way, the Plaintiff could appeal, and in practice frequently does.
A national candidate so challenged can even decide not to contest the State AG's decision, as did Abraham Lincoln in the case of the southern states in 1860, and simply forgo a place on that state's ballot. If he decided to contest the issue, he would appeal up the ladder until it reached the SCOTUS. Because of time constraints, it would be fast-tracked.
IMNVHO, the key to the failure of those bringing suit against Obama in recent cases lay in making him the "Defendant." IOW, the "Plaintiffs" had no "authority" to challenge; "no standing." Or, they were unable to show "damages," no tort against them.
Also remember that two Supreme Court Justices have candidly admitted that they have "shoved this one under the rug," turning away several appeals that had legal merit on very narrow legal grounds.
Bottom line. You are correct, vetting the candidates on a state ballot is quintessentially a state matter and a major key to having the case heard at the SCOTUS.
I guarantee there is no provision for your AG to interfere with the careful political considerations that are contained there.
If every state AG could do as you describe, opponents could drag out the process such that it would shut down the presidential election process. That is not going to happen.
We can look to actual, real life events that happened in Arizona as an example.
1) On November 30, 2007, Barack Obama signed a notarized Arizona Presidential Preference Election Candidate Nomination Paper attesting that he is a “Natural Born Citizen of the United States.”
http://www.scribd.com/mobile/doc/11107727
2) With that attestation, Obama was cleared for the 2008 Arizona Primary and the 2008 General Election Ballot in accordance with Arizona election statutes.
3) The Arizona legislature passed a bill that would require candidates to present evidence of their eligibility before being placed on the ballot.
4) Arizona Governor Jan Brewer vetoed the proof of eligibility bill.
5) John McCain won Arizona’s 2008 Electoral votes.
6) Arizona Tea Party members request that Arizona Secretary of State Ken Bennett get documentation of Obama’s eligibility from Hawaii before placing him on the 2012 Arizona ballot.
7) Bennett writes to the Hawaii Registrar of Vital Staistics and secures a Letter of Verification of Birth in Lieu of Certified Copy from Hawaii.
http://www.azcentral.com/12news/Obama-Verification.pdf
8) With the Letter of Verification, Bennett clears Obama for the 2012 Arizona ballot.
9) An Arizona citizen registered voter, Kenneth Allen initiates a ballot challenge in state court in Tucson to Obama’s natural born citizen status and seeks to bar Obama from the 2012 Arizona ballot.
10) The Court rules that Obama is a natural born citizen and is eligible.
Allen v. Obama, Arizona Superior Court Judge Richard E. Gordon: “Arizona courts are bound by United States Supreme Court precedent in construing the United States Constitution, and this precedent fully supports that President Obama is a natural born citizen under the Constitution and thus qualified to hold the office of President. Contrary to Plaintiff’s assertion, Minor v. Happersett, 88 U.S. 162 (1874), does not hold otherwise.”—Pima County Superior Court, Tuscon, Arizona, March 7, 2012
http://www.scribd.com/doc/84531299/AZ-2012-03-07-Allen-v-Obama-C20121317-ORDER-Dismissing-Complaint
11) Mitt Romney defeats Barack Obama in Arizona and wins Arizona’s electors.