The website that found that link ("What's your evidence?") says they received verbal confirmation from the newspaper that they had a similar policy back in 1961.
http://tesibria.typepad.com/whats_your_evidence/2008/09/reports-that-ob.html
The same newspaper also published an article in which they confirm that back in the 1960's, they published birth announcements on information filed at the Hawaii departmentof health. The link is in one of my posts above.
I don't see how Hawaii being a new state at the time is relevant to anything. Before it was a state, it had been a territory for a long time, and territories have governments with things like public health departments as well.
Again not good enough. Show me something in writing from (microfilm) the newspapers back in 1961 not from some obscure blogger who claims that he got "verbal confirmation." And I doubt the news people who put birth announcements in their newspapers were not concerned of using their birth section as an official Hawaiian birth count.