First of all, most of the lawsuits filed against him don't have anything to do with the birth certificate. There are the lawsuits claiming he's ineligible because he was a dual citizen at birth. There are other lawsuits claiming he in ineligible because a natural born citizen needs two citizen parents, and his father wasn't a citizen. Finally, there are the lawsuits claiming he lost his citizenship while in Indonesia.
Even the lawsuits related to the birth certificate wouldn't go away. Suppose he presented a judge with a birth certificate. The lunatics would come back with another lawsuit claiming the certificate is a forgery. Or they'd claim it's not sufficient to prove his natural born status because Hawaii's handed out BC's without sufficient proof. Barring that, they'd find another excuse to sue.
The idea that everything would just "go away" if he only provided a $15 copy of a BC to a court is no less laughable than any other birther claim.
Why on earth SHOULDN’T these constitutional questions as to “natural born citizen” be legally determined?
Is a child of a foreign national a natural born citizen under the U.S. constitution? That is a legal question and it should be determined in a legal manner.
You seem to operate under the feeling that Obama is akin to something like a king or a noble with a higher status than other individuals in the United States, and that he should be allowed to be untroubled by constitutional questions as to his eligibility.
He is not. Legally he has no higher status than any other person, and if he is not constitutionally eligible, then too bad, regardless of how inconvenient that may be to him.
He is, or should be, as an individual, subject to the same requirements of the constitution and laws of this country as any other person.