The crux of the officer argument comes down to these sections.
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No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been encreased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.
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The Senate shall chuse their other Officers, and also a President pro tempore, in the Absence of the Vice President, or when he shall exercise the Office of President of the United States.
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No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may by a vote of two-thirds of each House, remove such disability.
So a senator or representative cannot hold another office, but the president pro tempore is specifically an office other than senate or representative but has to be held by them. Additionally, the president is described using the same language of just being an office. So we are stuck to either say that "office" is not a rigorously defined term in the whole of the constitution (the obvious answer) or that "office" is a term of art that has to be interpreted differently based on the specific passage in which it is used and actually every called out office holds a special arbitrary place as "office", "office of the united states", and "officer under the authority of the united states".
As the latter requires doing backflips to maintain consistency in the constitution, we obviously have to take the former, and conclude that the 14th amendment then clearly would apply to the presidency as it makes no sense for it to apply to the vice president but not the president.