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.
For this whole born-in Kenya thing to work, here's how it would have had to come together:
Against all sense, Obama's 9-month pregnant, teenage mother flew halfway around the world to Kenya (a very difficult flight even today, never mind 1961) for some inexplicable reason. Then, a couple of days after the birth, she flew all the way back again, with a newborn, and they came back quickly enough that none of her friends figured out that Barrack was born in Kenya (ask any mother how likely she would have been to take such a trip a couple of days after giving birth). At the same time, her mother ran a birth announcement in the local papers because, with a daughter halfway around the world with a newborn, this was somehow on her priorities list.
His mother was able to bring the kid back into the US with no passport or any other identification for him and subsequently obtain a Hawaii birth certificate, somehow.
Then, for the next 40+ years, everyone with knowledge of this caper kept their mouths shut on the off chance that Barrack might some day run for President.
Does this story really sound within the realm of possibility to anyone?