I think may would say he is not eligible because he is not a natural born citizen, as is required by the constitution.
“No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President”
The constitution does not define natural born but most take that to mean born in this contry of two parents who are citizens. BO does not meet that definition. SCOTUS should take up the case just to clear up and define this.
Actually, what little case law there is on this leads most people to believe that those born in the United States, regardless of parents are natural born citizens. Its a valid point of argument and you are correct that SCOTUS has never directly ruled on this clause as it pertains to the Presidency. I suspect that they don’t want to do so either, preferring to leave this sticky little problem to the politicians who will bob and weave as they are wont to do.
Most do not take that to mean what you say, and the only court case in all this to consider that question in it's opinion came to the answer “born here.” That wasn't what they were asked to determine, but they made their opinion and associated reasoning clear and it has not been successfully challenged.