"Leading the nation" is completely different from being Commander in Chief (CINC) of the Army and Navy. One is a specific designation of military authority, the other is rather more nebulous and touchy-feely. Being CINC means that the members of the armed services are obliged to obey his lawful orders. As I (to take an example) am no longer a member of the military, I am not obliged to obey his orders. Many in the media, especially during the time immediately after September 11th, mistakenly referred to the president as the CINC of the nation, or of the United States. This is really only true if we become a military dictatorship.
Regarding the Presidential Oath of Office: It is not simply an oath or affirmation, it is The oath or affirmation. It is of such importance in defining the duties of the president, that when the House Judiciary Committee passed articles of impeachment against Richard Nixon, the articles specifically cited violation of the oath as justification for removing Nixon from office.