I'll also add that taking the oath isn't even necessary. For example, LBJ became president the moment JFK was pronounced dead, not when
this picture was taken about two hours later. Or for another example, there is a long-standing myth that
David Rice Atchison, the next in line to the presidency in 1849, became president for one day when Zachary Taylor refused to swear in until the next day after the designated inauguration day. Oath or no oath, Taylor became president the moment Polk's term expired, and Atchison was never president.