That makes literally zero sense, because the 14th Amendment bars anyone from holding any civil or military office who engaged in insurrection. And before you go on about "well durrrrr the presidency isn't an office," the constitution refers to the presidency as the Office of the President of the United States repeatedly:
Article 1, Section 3:
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.
Article 1, section 3:
Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.
(This provision is especially important because it means that if the presidency isn't counted as an office the president is literally immune from impeachment because there's no provision in the constitution to actually try the president for impeachment.)
Article 2, section 1:
The executive Power shall be vested in a President of the United States of America. He shall hold his Office during the Term of four Years, and, together with the Vice President, chosen for the same Term, be elected, as follows...
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected...
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."
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.
12th Amendment:
But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States. *Superseded by section 3 of the 20th amendment.
22nd Amendment:
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.
In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
I could go on, but I think you get the point. Claiming that the 14th Amendment doesn't cover the presidency completely ignores the plain text of the entire fucking constitution!
Another fun side effect is that the president not being an office covered under the 14th amendment would also mean they're exempt from the Emoluments clause, though that one might already be dead.......