As a matter of intellectual curiosity, perhaps.
As a constitutional matter, the issue was closed when the special Joint Session of Congress convened for the purpose of, among other things, hearing objections to certifying the election, heard none.
As far as the Constitution is concerned, the matter was closed when the President of the Senate signed the papers.
Not so sure about that. Fruit of a tainted tree is still a potent legal theory in this country.
It is closed also as a practical matter because there is simply no energy behind the cause. It is closed as a juridical matter because I cannot conceive of a scenario in which the court will not sidestep this matter as a political question or as a question more properly determined by the legislative branch under the Constitution, as you point out.
There is however a faint hope that enough states might pass legislation requiring production of documents cause Obama some difficulty in his quest for reelection. This is however only the faintest of hope.
The matter is effectively dead without dramatic, extrinsic evidence of his birth outside of the states. But this is a wholly different matter and the facts do not warrant denigrating the whole of the "birther" movement as crazy or racist. But even that reservation is a losing argument in the battleground of popular ideas.
So, the President of the Senate can give exemptions to Constitutional requirements?
I don't think so. I hope to hell not anyway.