You might be right, but I doubt it. Taking the oath of office is a requirement for taking office, but it does not make the person being sworn President. It's just another requirement for becoming President on January 20th, at noon. Just like being eligible and getting a majority of electoral votes. But, given that the requirements are all satisfied, it's the clock going past noon January 20th, that makes one President. Specifically the Constitution says:
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:
and
The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
Note that it does not say that the President elect become President upon taking the oath, nor even just when before entering the execution of the office the oath must be taken. (I'll not even get into the oath fiasco. :) ). It's only by tradition that the ceremony includes the oath takings is done on right at noon on the 20th, or in the date in March that the Presidential term began before the XXth amendment changed it to January 20th.