If what you suggest is true, why did Obama scramble so fast to produce that multi-layered LFBC monstrosity the instant Trump started championing the ‘birther’ issue? If ‘birtherism’ was really good for Obama, shouldn’t he have ignored Trump & allowed him to go hog wild? Yet the moment Trump began shining a spotlight on the issue, voila, out of the woodwork comes a LF BC, paraded out in front of the cameras for all the world to see. Despite the obvious cut & paste/computer-generated nature of the doc, it laid the issue to rest for many. If Obama profited from the issue, why did he want it to go away?
Go away? How could it go away?
Trump was using the issue to gain traction in Republican polls. Others were using it as a basis for fundraising (i.e., making money). People were financially and politically benefiting from the issue. Why would any of them allow it to just go away? Everyone knew that there wasn't then and there isn't now any way to prove with absolute certainty where Obama was born or where anyone else his age was born for that matter.
Obama had to know that releasing more documents would not make the issue go away for people who had become dependent on the issue's existence. And, he was right. It didn't make the issue go away. What the release did do is make birthers appear (to most people) to be just that much more nutty and kooky.
If birthers were provided a courtroom to try this case tomorrow, who would appear as a witness to competently testify that Obama was born in Kenya or anywhere in the world outside the United States? There isn't anyone willing and able to do that. The problem with the case is that there is no case. There is fundraising, there are book sales and there are faux lawyers filing faux lawsuits, but there is no underlying case.
Hell, even the birthers who now believe that Obama may have been born in Hawaii haven't come home. Lots of them just shifted their focus away from his place of birth to the citizenship of the man they believe (but can't prove) was his father. In order to do that, they have gone so far as to imply that the ordinary citizens of our constitutional era were more or less uniformly conscious of and in agreement with the theories of citizenship written in another language by some Swiss philosopher named Vattel. All of it is of course baloney, but to the people who had wandered that far into the intellectual jungle that they created, nothing could ever make the birther issue "go away." Obama had to know that much. And, as you can see, he was right.