...open to the public with the permission of the authorities.
No, it isn't. You consistently invent non-existing procedures and policies in order to make it seem that "investigators" are being pointedly rejected or stymied in their efforts. In fact birther "investigators" are simply running up against the ordinary laws that apply to everyone. Birthers are not entitled to special exemptions from state and federal laws, nor authorized to invent new
ad hoc procedures for determining eligibility. The imagined importance of their hobby is irrelevant.
But Sheriff Joe's investigators were refused admission to the "public" archives.
No. Hospital admission records are protected by HIPAA. Federal law governs whom they can be shown to and why. There are criminal sanctions attached to even the accidental release of that information to unauthorized parties. Hospital administrators, law enforcement officials, etc.
cannot simply decide to ignore those laws. There is only one person in the world with the authority to grant permission, and that's Barack Obama's mother.
Had Arpaio's investigators obtained a search warrant, they would have been able to avoid HIPAA restrictions. But two things prevented that: first, that Arpaio's evidence has already been ruled insufficient for prosecution, thereby ensuring that no state or federal judge will grant a search warrant to continue to pursue the matter; and second, that Arpaio's investigators are now enjoined from acting under color of law while investigating Obama and are thus ineligible to petition for a warrant.
The investigators originally tried to get around the latter by saying nebulously that they were "from the sheriff's department," but their lie was quickly discovered and they were shown the door. In that circumstance they were not law enforcement officers acting in their official capacity with due jurisdiction and authority, but were instead private citizens requesting access to information that long-standing federal law prevents them from seeing.