Do not conflate “at birth” and “by birth”.
“At” is a point in time.
“By” is causal.
§ 1401 declares who shall be nationals and citizens of United States AT birth. Such persons are citizen at birth by law.
The causal agent is operation of law. That law may vary causing a person to either be a citizen at birth or not at all.
A person born in country to citizen parent is a citizen BY birth.
The causal agent is birth. This person is always a citizen, this has never been doubted.
Again, I would challenge you to find one single court decision in the entire history of the United States that makes a legal distinction between the two.
I would challenge you to find one authoritative legal opinion in the entire history of the United States that makes a legal distinction.
And I would challenge you to find one single statute in the entire history of the United States that makes a legal distinction.