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 provide 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.
There was the consideration during the amendment processes that we may have a case where a person was 'elected' but failed to qualify. In other words, something like what may be happening here wasn't out of the realm of possibility.
What you quoted almost gives it structure. Sounds like the time for discovery about him being eligible or not is between the time the electors vote and the inauguration.
Of course, even after the inauguration, if he is proved to be ineligible, he’s outa there in a NY minute.