You carry your American citizenship with you wherever you go. If an American citizen creates a child in another country, the American citizen can go, if she/he wants to the American Embassy and register that child as an American citizen.
My son married in England to an English woman. They lived in London, child born in London and son went immediately that day to the American Embassy and the son was an American citizen that day and an American Passport for the new American citizen was given to son that very day. He has dual citizenship, American and British.
When the child comes to America, he goes to the American citizen line to come into the country. His mother, being a British citizen, has to go to the out of country line to come into the country.
You are absolutely right. However, what we are debating here is the special designation that becoming POTUS carries with it, that being the term “natural born citizen” but many do not make the distinction. Some say your grandchild could be president and others do not.
You are not a US citizen if you have dual citizenship. To be a US citizen means you own allegiance to the US and only the US. For most of our history that was the law, and I hope once again to be the law. The Marxist left changed that and our migration laws back in LBJ days. We need to change those laws back to pre LBJ days.
I can't figure out why people believe that our Founders, who were concerned about the loyalty of the President above and beyond the loyalty of any other official, would find it sufficient if the President had just a single citizen parent rather than both; as if loyalty to that one non-citizen parent does not create a conflict of interest.
European monarchs routinely found themselves married to foreigners for the very purpose of creating alliances which their children would recognize and by which they would be influenced.