There are Officers not appointed by the President. For example, the Speaker of the House: "The House of Representatives shall chuse their Speaker and other Officers".

Sure - but the Speaker isn't an "Officer of the United States." Because those Offices can only be filled by Presidential appointment.

So when you look at the DQ Clause of the 14th, it provides:

"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...."

...which doesn't mean any single position that is called or can be called an "office." It refers to offices "under the United States." That term (like so many in the Constitution) is undefined. Since the Appointments Clause says that all other "Officers of the United States" are to be appointed by the President, though, there's a strong argument that "Officers of the United States" means the ones subject to Presidential appointment power (which doesn't include the President).

And again, the fact that not every position mentioned in the Constitution is an "officer" for the purpose of the DQ Clause is reinforced by the fact that the DQ Clause specifically mentions Senators, Reps, and electors in addition to "any office." All of those folks are presumably not holders of "any office....under the United States." Which means that there can be important members of the Federal government which aren't officers of the Federal government.