That "ruling" makes no sense. Wong Kim Ark's father was a permanently domiciled alien in the U.S. at the time of his son's birth, and the ruling in U.S. v. Wong Kim Ark was based on that and other agreed upon facts of the case.
Obama's father WASN'T a permanently domiciled alien at the time of Obama's birth. U.S. v. Wong Kim Ark doesn't even apply to Obama.
Immigrants, undocumented immigrants, and legal permanent residents are required to register.
Men who are in the U.S. on student or visitor visas, also referred to as transient aliens, aren't required to register.
Obama's father was never an immigrant, undocumented immigrant, or legal permanent resident. He was always in the U.S. on a "temporary stay" (exact DoJ quote at the time) as a foreign student.
U.S. Secretaries of State have already ruled that children born in the U.S. to such transient alien fathers weren't automatically U.S. citizens at birth because they were born 'subject to a foreign power,' and therefore not "subject to the jurisdiction of the United States" within the meaning of the Fourteenth Amendment.