What you propose is that one provision of the Constitution modify another provision.
Such a construction is impermissible.
The 14th Amendment made former slaves who were not Citizens in any sense of the word “Citizens”. Was it the intention of the drafters that these new Citizens were to be considered anything less than Natural Born Citizens?
They were not born citizens and they were not the children of citizens.
So were any of them eligible for president after the 14th Amendment?