The oath of office is not issued, like a King's scepter or crown, it's taken by the President-Elect. Taking it is required before exercising the powers of the office, but taking it does not a President make. Getting a Majority of the electoral votes, being eligible, taking the oath and then the clock and calender, a President make. Except in the case of death, disability or removal of a President, in which case it's that event, plus being eligible plus taking the oath. Generally taking the oath is the last "wicket" in both instances, but all the wickets must be passed before one is President. If you don't meet *all* the wickets, you aren't President, and have no power to act as such.
Also, the oath need not be administered by the Chief Justice, and many times has not been. Most of those were early on, or were the cases of Vice Presidential succession due to death of the President. The big ceremony with the Chief Justice is *tradition* and *cusyom* not law. It's certainly not a Constitutonal Requirement.
Being over 35, 14 years a resident, and a Natural Born Citizen is though.
Yes, you have stated the process correctly as it is delineated in Article II of the Constitution.
Article II Section I Clause VIII: “Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:—’’I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.’’