Rogers v. Bellei covered the special case of a child born abroad to a single US Citizen parent.
Why did you go to the consular officer in the US Embassy, Moscow?
For the (quite practical) reason that the children need US Passports. Again, the Embassy did not naturalize the children. The officer simply reviewed the (Russian) birth certificates (of the children) as well as our (parents) US Passports, and provided a document that states that "yes, these children are Americans", and subsequently accepted the forms for their passports. The document provided (CRBA) can be used in place of a US-state-issued birth certificate for any and all purposes.
Natural born citizenship has no need for resort to any consular's office.
If you believe this, I would suggest that you attempt to apply for a passport without any evidence of date/location of birth and/or parentage. You will not get very far.
In a pinch, a birth certificate gets the job done.
In your closing argument or statement, you are confating fact of citizenship (can't get a passport without evidence of citizenship) with source of citizenship. Bellei (and your child) were made US citizens SOLELY by dint of statutory law.
If the statute didn't exist, if there was no US embassy in Russia, would your child be a natural born US citizen? You say yes. I say no.
I'm not going to discuss this further with you, my mind is made up, and so is yours. We will never agree on this subject. You get the last word.