Another question about the original scenario. The vice-president becomes president as soon as the president dies. Is he included on the list of presidents even though he is never sworn in?
David Rice Atchison was President for 1 day in 1849 because Zachary Taylor refused to be sworn in on a Sunday and he is never mentioned as a President.
By your reasoning it is required that a President swear in to actually become President: Since Taylor did not swear in on the Sunday, he was not President. Since Atchison did not swear in either, does it not stand to reason that he was not President either?
Arguably, the United States did not have a President at all for that day since no person was duly sworn into be President.