Out-of-wedlock births were hidden in those days..
Since the parents at least claimed to be married the mother would not have had to live in a home for unmarried pregnant teens and no one would necessarily have thought it was an out-of-wedlock birth.
Since when are marriage certificates private documents? This should be on record with the County Clerk’s office.
Maybe there wasn’t a wedding. Obama Sr was married already in Kenya and probably didn’t want to break the law and commit adultery. He had high ambitions at this point.