To me it means nothing; the US does not care what another nation claims on you, only what you claim for yours. If another nation claims you as their citizen, and you do not recognize that claim, then that is good enough. The "dual citizenship" is a non-issue because President Obama never claimed it as his own.
There remains, however, something that will toss him out and that is any proof that he was born outside the USA
On that I completely agree, and is the only reason I'd want to see the birth certificate. It has nothing to do with his father's citizenship or even the fact it was an illegal marriage (not recognized). If he wasn't born here then - by either Vattel OR English common law - he's not a natural born citizen.
“To me it means nothing; the US does not care what another nation claims on you, only what you claim for yours. If another nation claims you as their citizen, and you do not recognize that claim, then that is good enough.”
“...the US does not care...”?
You appear to have made this up as the founders certainly did care. See John Jay’s letter to Washington which appears to have directly resulted in the NBC clause in the Constitution.
FWIW I think the Mombasa claim began as a political canard invented by Obama himself, to help overcome the "blackness deficit" identified in his ill-fated run against former Black Panther Bobby Rush. I think he kept up the "I was born in Africa" roorback from the 90's sometime, until 2006 when he decided to run for President, and someone realized the trap.
I think his real problem is in his passport use in his teens and 20's -- he may have been a foreign/dual citizen. If he had an Indonesian passport, then he was claiming Indonesian citizenship, because Indonesia disallows dual citizenship.