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.
"the Vice President elect shall act as President until a President shall have qualified" sounds to me like a new election. I can't imagine any party agreeing to a losing party being declared the winner.
You’re right, the succession would go to the VP-Elect and then to the other pres candidate if the VP-elect failed to “qualify”.