“ prevent any American born American citizen from running for president.”
There’s nothing in the Constitution about “running for President”. That’s because there’s nothing in the Constitution about Presidential elections.
If the NBC clause were ever to be used, it would have to bar an inauguration of a person found ineligible under that clause.
Article II, Section 1, Clause 5:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
Correct, the Constitution does not mention “running for president” in the discussion of who is eligible. It indicates “eligibility” for being President.
But elsewhere, the Constitution discusses presidential electors, which touches on the election of a president through the actions of the electors.