anyone can run a birth notice
Yes, dalebert, of course you're right. But there's such a thing as commonsense. And commonsense tells us that a contemporaneous notice in the paper in August of 1961 is more likely than not to represent the truth: The Obamas of Hawaii had a boy on August 4.
Yes, that notice could have been run by anyone, but commonsense tells us that it was likely the Obamas or a close friend or relative in Hawaii who ran the notice.
Now, the birth notice does not say the baby was born at a hospital in Hawaii. (Nor do the adjacent notices in the paper, for that matter.) In fairness to the conspiracy crowd (of which I'm not a member), that leaves room to argue that the birth could have occurred elsewhere, say in Kenya.