The Constitution includes no mechanism for determining the citizenship status of a candidate for a very simple reason.
The original Constitution did not have the people voting for President at all. They instead voted for members of the Electoral College which would then elect the president. The EC would be a small elite group that would be quite capable of determining qualifications for itself. The Convention did not want to tie the hands of the EC by providing rules for its operation that would be unnecessary in any case.
The EC never once really functioned as intended. Everybody knew Washington would be elected by acclamation, which he was, twice.
By the time of the 3rd election candidates for EC were running while openly pledging themselves to vote for a particular candidate, which eliminates most of the rationale for having an EC.
Article. [XX.] [Proposed 1932; Ratified 1933]
Section. 3. If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provided for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.