The founders clearly established three kinds of citizens and only one was eligible for the presidency, the Natural Born Citizen, the other two were only eligible to be Congressmen.
They did not put things with no validity in the constitution and meant what they said.
In the 50s and 60s it wasn’t even discussed in the vast majority of civics classes and was never a real issue. My hs education was much superior to the crap they push today and actually had truth in it. I feel privileged.
The information you discuss can be extrapolated from the birth certificate if the parents names as on it. But County Clerks are not charged with investigating any further than the unsworn statements.
secretaries of state are responsible for vetting the people they out on the ballot. When Eldridge Cleever ran for president on the Peace and freedom ticket, the sec of state of Cal asked for his birth certificate. He was not old enough to run and was left off the ballot.