Do you imagine a 50+ year plot? Doesn't pass the smell test. Easily doesn't mean likely or even probable.
No conspiracy shit needed, either.
Let's say my mom and dad had gone back to the old land while she was heavy with child (moi). And just before she was set to fly back, they discovered that flying increases the risk of complications (yada yada). So the airline wouldn't let her fly. Hoo boy. She tries to arrange alternative transportation but sneaky old mother nature says, "ha!" So, there and then, I am born in another land.
Now mom isn't thinking about me running for president somewhere 35 or 50 years down the road. She just figures it will be a freakin' hassle for me to have to explain for the rest of my life that I wasn't born here. (As luck would have it, my dad is an alien to America and I was born in HIS homeland for purposes of this brilliant hypothetical, so don't get all worked up.) In fact, they want to make sure that I am always considered a U.S. citizen -- and JUST a U.S. citizen. So, she and dad sneakily CLAIM that I was born in NYC! Cut right the hell through all that Immigration Law crap and red tape and any bothersome questions now and for the rest of my days. (And even though I was "there" at the time I was born, damn'dest thing; I just don't remember it!)
Anyway, as part of the clever mom and pop cover story, they place an ad in the local newspaper announcing how I was born in that hospital in the Big City! The paper doesn't check it for validity. They just collect the ad fee and print that shit.
Years and years later, if I were to run for some big old elective office, maybe parts of my life would get vetted. And since the press would love my highly liberal ass (this is an alternate universe), they wouldn't REALLY vet me all that much. Cover story remains intact. Questions? Yeah. Maybe. Dismiss that as the looney musings of those conservatives and birfers. Who cares? I am a liberal so the press is at my beck and call. No need for that "investigative journalism" stuff.
And of course, I go on to kill Osama and ruin the economy. So all is good.