Not only is what you're saying obviously correct, but THE PEOPLE WHO VOTED IN THE FUCKING AMENDMENT was asked about this exact lack of the word "president", and pointed to the officer line, and said that clearly included the president.
It's also worth noting that there was just a single reference in the Senate debate to the fact that the president and vice president were not explicitly mentioned in Howard's draft as "officer(s) of the United States," the way members of Congress and state officials had been itemized in the text. Would the disqualification clause of the amendment not cover the top posts in the executive branch?
"Why did you omit to exclude them?" asked Maryland Democratic Sen. Reverdy Johnson.
Maine's Lot Morrill jumped in to clarify.
"Let me call the Senator's attention to the words 'or hold any office, civil or military, under the United States,'" Morrill said, ending the discussion on that point.
It's so infuriating how difficult it is for people to understand this