Exactly. Nowhere does it say he must already BE President to take the oath, or that the taking of the oath CAUSES him to become President. It is the arrival of Noon that does that.
Correct me if I’m wrong, but not having taken or having on file an oath of office nullifies any official actions taken by public servants. IIRC there was a stinkfest a few years back in LA over the fact that a bunch of court officials didn’t have them and it was going to have a bunch of prosecutions thrown out.