Article III of the US Constitution says the judiciary is given the responsibility of deciding any cases resulting from the US Constitution or the law.
The US Constitution says that there are certain requirements for anybody who is even ELIGIBLE to be POTUS. If there is a case arising out of the Constitution’s requirements for eligibility, the correct mode for people to “petition the government for a redress of grievances” (a First Amendment right) to decide that CASE is the judiciary, which includes SCOTUS.
Congress has no power to decide legal cases. They are totally irrelevant to this discussion.
But the process to put someone in the Presidency is political from beginning to end, deliberately, by design.
You are correct that the USSC may decide "cases arising under this Constitution". But the USSC has never defined this question as a "case", and, in my opinion, they will not (and should not) do so.
The courts do not decide political questions. The whole elaborate ritual of opening and counting the sealed votes in front of the sitting VP and every Senator and Member of Congress was designed to emphasize the entirely political nature of the process.
And, lest you think I am defending "Obama" or his interests, I am not.
That a person such as the man who calls himself "Obama" could have been chosen to be President is a symptom of very, very deep-seated problems, problems which me may well not survive.
The Supreme Court, however, has already amassed (much) too much power. They should not be accorded the power to dismiss a President.
That's our job in the first instance (by not electing him) and Congress' job secondarily.
Should it be necessary to remove him by extraconstitutional or even extralegal means, the harm tha would be visited on this land would be, in my view, extreme. I pray that that will not become necessary.
It is my view that the USSC is correct is avoiding this, the ultimate political question. We have to resolve it ourselves, or live with the consequences.