And yet, this is not borne out in reality: what happens if you are a soldier who refuses to abide by orders due to the superiors having no authority? {The chain of command is dependent upon the fundamental qualities of authority: the one who is sent is under the authority of the one sending; this goes to the Commander-In-Chief, which the Constitution assigns as the President.}
However, of those people who do care to know who is president of the United States, probably 99.99% of them believe that Barack Obama is the president of the United States. By showing up for Obama's inauguration and by showing up for Obama's state of the union speech, most members of the Supreme Court are signaling to you that they believe that Obama is the president of the United States. The Chief Justice administered Obama's oath of office.
Like that vast majority of Americans, the Supreme Court believes that Barack Obama is the president of the United States. They may not be absolutely certain that Obama or any of our past presidents met all of the qualifications to be president or should have been president, but they do believe that all of these people were in fact presidents. They don't see anything in the Constitution that empowers them to either raise the question or attempt to answer the question as to whether any of these folks should have been president or met all of the qualifications. All of our presidents became president as the result of a political process, not a judicial process. The people and their electors decide upon the candidates' qualifications.
You don't need to accept that you believe that Obama is the president for you to accept that the Supreme Court believes that Obama is the president. The Supreme Court does believe that somehow Obama became and now is the president of the United States. That's all they need to believe and know that only the Congress can remove him from office.
And that's what the courts have been trying to tell Orly and her fans.