You need to acquaint yourself with the concept of Judicial Review in the United States.
I am claiming that the U.S. Constitution delegates powers to many persons and entities. A local jury in a criminal case can decide in a criminal case that the facts do not warrant a conviction no matter how strong the prosecution's evidence and no matter what any judge might think the jury should have found. That may seem unfair to you, but that is our system. The Congress can declare war against another nation even if all the judges think that the decision involved a misunderstanding of the facts by the Congress. That is our system. The President can veto a law even if every judge in the country thinks he is doing so because he is mistaken as to some important facts. Again, that is our system.
You seem to be concerned that the voters and their electors might make a factual mistake in deciding whether or not a candidate meets constitutional qualifications. Of course, they might make a mistake. Everyone who is granted the power to make a decision in our country might make a mistake. People might disagree as to what the facts are and what the facts mean, but that does not mean that every decision is reviewable by a court. That's just our system.
The Supreme Court is not granted any power by the Constitution to pick our presidents. The voters and their electors are quite capable of resolving questions concerning qualifications.
Do you at times wonder why the Supreme Court has not accepted any invitations to rule on the factual questions concerning Obama's qualifications? Well, now you know why.