He could, but he hasn't, and IMHO, he can't. Because if it becomes legally recognized, then the "Born of Citizen parents in the US (plural) test becomes a possibility. Even the English Common law definition had the natural born citizenship of the child going with the citizenship of the father, in cases where the child was born outside the father's country.
The birth certificate is one piece of evidence leading to legal recognition that his father was not a citizen. But as you say, he could stipulate it. I don't think he will. Not stipulating it if asked, could bring a heavy political penalty, but not a legal one.
But of course if the birth certificate shows circumstances that he might not have been born in the country and that the certificate may have been fraudulently filed, such as a certificate that indicates a late filing or a filing of a home birth, then the Kenya BC would into play and more discovery would be required.
I was sloppy. He could admit it in his pleadings once we get past discovery.
Everyone wants him to plead facts in his pre-trial motions. This cannot be done.