Read the whole page. It clearly explains what the program is, and what the requirements are. There's no mystery about what they need to verify. They are verifying ancestry, not just Hawaiian birth. The "further verification" is the verification beyond your own birth, your ancestry.
"Spin on though you are amusing me with your convoluted concatenations."
The convolution is your own. That site is very clear about the program and its requirements. You guys are trying to twist anything you can into supporting an idea you've already decided to believe in. That is spin.
Even with the Certificate of Live Birth the Department still wants to see certified copies of your parents' CoLB's and even grandparents' CoLB's to independently verify where they were actually born.
That is what the extra work and charges involved with the Certification are about; to fully corroborate that you actually were born in Hawaii (for starters). The Certificate lists the name of the hospital, the address of the hospital, the name of the physician that performed the delivery, etc.