“A child born abroad to one U.S. citizen parent and one alien parent acquires U.S. citizenship at birth under Section 301(g) of the INA provided ...”
But only after securing U.S. diplomatic recognition of the U.S. citizenship after the occurrence of the birth.
That doesn't make logical sense, and even if it did, it omits obtaining certification via U.S. Citizenship and Immigration Services, which is not the same as diplomatic recognition.
The statute says what it says, "shall be nationals and citizens of the United States at birth", and the USCIS webpage says,
Form N-600, Application for Certificate of Citizenship, is filed to obtain a Certificate of Citizenship which serves as evidence of your or your child's U.S. citizenship. You may file Form N-600 if you were born abroad and are claiming U.S. citizenship at birth through your parents. You may also file Form N-600 to obtain evidence of citizenship if you automatically became a U.S. citizen by operation of law after your birth but before you turned 18 years of age.USCIS FAQ Page
If you and kabar are correct, that paragraph would read remarkably differently. Perhaps along the lines of:
Form N-600, Application for Citizenship, is filed to obtain a Certificate of Citizenship which creates your or your child's U.S. citizenship. You may file Form N-600 if you were born abroad and are seeking U.S. citizenship at birth through your parents.
Under your and kabar's view, the USCIS sentence about citizenship being automatic for some foreign born persons is impossible. No foreign born person automatically becomes a US citizen by operation of law.