Correct. All I can say is that I worked at a small town paper for quite awhile, and the same info from the same government agencies were printed regularly, week after week, year after year -- they filled space between ads, and we counted on them. So I can believe that the notices came from the DOH in 1961, as they did years later.
But your belief mileage may vary.
You are missing the point about the birth announcement appearing in the papers. The mere registration of the birth with the Department of Health would generate the birth announcement, whether the birth happened in a hospital or elsewhere and it could have been reported by Madelyn Dunham!
All I am saying is that currently newspapers print birth and other like notices from info given by actual people as well as from state, city or county departments. People have posted on FR dozens of times how they or their relatives phoned or mailed in birth info and it was posted in the home town or whatever newspaper. To this day it can be done like that in many places.
I lived in HI less than ten years after 0bama was born and the atmosphere was still very laid back and small town. Until proven otherwise, I think it more than likely that people could give the newspaper birth info (or marriage, for that matter) and the papers would print it, as well as offiical info from the HI Heath Dept or whatever it was.
Now I don’t know this for sure, but your assertion that the announcements were without doubt generated solely by the state dept is not proveable one way or another. Maybe, maybe not. And even if that is so, it still doesn’t prove where 0bama was born, just that some kind of birth record had been recorded by (one would assume) his mother.