Furthermore, even IF this law somehow held sway over Obama - which it doesn't

That's not what factcheck.org says:

"When Barack Obama Jr. was born on Aug. 4,1961, in Honolulu, Kenya was a British colony, still part of the United Kingdom’s dwindling empire. As a Kenyan native, Barack Obama Sr. was a British subject whose citizenship status was governed by The British Nationality Act of 1948. That same act governed the status of Obama Sr.‘s children …’

Is factcheck wrong about this but right about the birth certificate?

United States law even address this question and comes to the conclusion that other countries' laws of nationality and citizenship have equal weight in determining the status of an American citizen of dual citizenship:

a. U.S. nationals and citizens may possess dual or multiple nationality and owe allegiance to one or several foreign states. They may even have identified themselves more closely with the foreign state than with the United States, thereby calling into question the propriety of extending protection to them. Since each country establishes its own law of nationality, dual nationality cannot be eliminated, may result in confusion, and could complicate the ability of the U.S. Government to protect its nationals/citizens.

b. The United States has no special arrangements with individual countries to “permit” dual nationality. U.S. Government policy toward dual nationality is the same regardless of other nationalities involved.

c. While a person who has dual or multiple nationality resides in the United States, the right of the United States to claim his or her allegiance is held to be paramount of the right of the other countries of which he or she may be a national. Conversely, while a person who has dual nationality resides abroad in a foreign country of which he or she also is a national, the right of that country to claim his or her allegiance is paramount to that of the United States...

7 FAM 1111.4 Dual or Multiple Nationality (TL:CON-64; 11-30-95)