I believe Obama withholds the documentation for political gain alone. He can point at the "Birthers" as his own "vast right wing conspiracy" and diminish his opposition by dismissing them as kooks.
But let's say SCOTUS discovered the long form doesn't support Obama's qualification for the presidency. BOOM! Immediate constitutional crisis. Is anything he's done "legal," are the very members of the court he nominated valid? Legislation? Treaties? Appointments? It's a nightmare that SCOTUS would prefer to avoid.
They've been here before. They got into Clinton's mess by saying a sitting president can be used for wrongdoing done before he reached the office. It seemed reasonable and yet BOOM constitutional crisis because the SOB LIED under OATH! How could they have predicted it?
Then in 2000 they faced Bush vs. Gore. They had no choice but to sort that mess out but it tainted the perception of the court and undermined the Bush presidency from day 0. The "selected not elected" lie is not something he ever recovered from even after 9/11.
Further, they have to look closely at what "natural born citizen" means. I personally don't have faith SCOTUS would go for an interpretation that requires citizen parents these days.
It's all amplified by prospect of potentially removing the first (half) black president and perhaps doing so with a 7 member court.
“It’s a nightmare that SCOTUS would prefer to avoid.”
I know that’s the prevailing theory, I just question it. I think we have type A people on the Supreme Court who would love to make history.