Because we don't trust *him*. He needs to be removed from the "chain of custody" of any documentation. Plus the Birth Certificate, that he (supposedly) has released is not sufficient anyway to prove natural born citizenship. If he released a copy of his "long form" certificate, with raised seal, that would be the "original record", or at least a certified copy thereof, that would be sufficient, but he would still need, for maximum credibility, to be out of the "chain of custody".
Heck they have more concern for chain of custody of my vial of urine when I go in for my random drug test, required because I'm an employee of a company with US Government contracts.
I have, starting when I was a USAF officer, then as a contractor, been doing it for several decades now. They don't trust you at all. First they notify you a very short time before the sample is to be obtained. Then they make sure you aren't carrying a vial or bottle of "clean" urine, when you go in, and they grab the cup as soon as you open the door to the head/latrine/bathroom. You have to sign that it is yours as does the nurse who takes custody and sends it to the lab.
I really don't think it's asking much that BHO have the Hawaii Department of Health Statistics send a certified copy of his long form birth certificate to a court.
A birth certificate is considered legal proof of place of birth.
You don't trust him. Fine. You shouldn't. But that's why there's the added provision of the state confirming the birth certificate.