The President is a mere public servant, not a feudal king. Inauguration is like a ribbon cutting ceremony, which gives the politicians a chance to make speeches. In itself, it is meaningless.
Unlike the giving and taking of mutual oaths for a kingdom, there is a single oath for a President, and it is does not bind the President to any mortal, but instead to the Constitution. The President doesn’t exchange oaths with vassals which must have witnesses in order to avoid fraud.
The Constitution requires that prior to assuming office an Oath be taken, this is part of the eligibility requirements like age and citizenship. It does not specify who (or even if) else must be present or when the Oath must be taken. Given that there is an implicit “during my term of office” in the Oath, a two term president could easily argue that the first oath is still binding and doesn’t need to be renewed for a second term.
Having taken the oath, at the appointed time, with no ceremony required, the person becomes President and is able to (legally) exercise the power of the presidency.
To put it another way, Inauguration is like a victory lap in racing or a boxers hand being raised by the ref, it’s a way of showing off and make the fans happy, but isn’t in itself important. If the wrong driver takes the victory lap or the ref raises the wrong hand, that doesn’t change anything. Likewise if the President-Elect were to be unconsciously and undergoing surgery at noon, the office would still pass and the President would still be the President.
Or to put it corporate speak—all of the on boarding formalities have been duly executed, the Inauguration ceremony is just the walk around the office where the new hire gets introduced to everyone.