It doesn't specify executive branch of any state. "Of any state" applies to "judicial officer", not to "executive". If you ask me if I have a cat, or a dog with orange hair, the orange hair part of the question only applies to the dog.
If I were to ask "do you have a cat or a dog with orange hair" then I expect most people would
not interpret it the way you describe, and would instead naturally interpret the phrase to be asking if you have a dog with orange hair or a cat with orange hair. The phrase
could be interpreted in the way you describe, but it would not be the natural one and if that was the goal, one would typically reorder it to make it clear, by asking "Do you have a dog with orange hair or a cat?"
Now, I do notice you put a comma after the word cat, so perhaps your claim is supposed to be that the usage of the comma there differentiates the two. The problem is, that only goes against your interpretation. Note the Disqualification Clause again, with the applicable section bolded:
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.
There is no comma between "executive" and "or judicial officer of any State". So if that was your claim (I'm not sure if it was), then your argument doesn't work.
Can you point to any legal scholar--anywhere--that endorses the interpretation you have offered?
But regardless of whether any legal scholar takes that interpretation, we can see multiple problems why it makes little sense to try to read "an executive" as separate from "judicial officer of any State" (rather than it being a shortened form of "an executive officer of any State or a judicial officer of any State"),
First, simply stating "an executive" with no further qualification would be a very odd term to mean, as "executive" can have a ton of meanings, but "executive
officer of any State" is much more clear.
Second, every other item in the list becomes with "as a(n)". "
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". If "judicial officer of any State" was to be separate from "an executive" then why does it not say "as a" before it?
Third, if "executive" and "judicial officer of any State" are separate categories, then why is there not a comma between them? When making a list, you put a comma between the things in the list (or semicolons if the things in the list are full clauses). That's something people learn in elementary school. Now, some styles will omit the final comma--often called the Oxford comma or serial comma--in the list (e.g. instead of "A, B, C, or D" you write "A, B, C or D") but it is clear that is not the style being followed, as we see the serial comma used elsewhere in the Fourteenth Amendment. For example, Section 1 reads, "But when the right to vote at any election for the choice of electors for President and Vice-President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof..." Notice the usage of the comma before "or the members". So the style being used is to include a comma before the final "or" or "and" in a list. So your interpretation goes against the laws of grammar that other parts of the Fourteenth Amendment are clearly following. Actually, the phrase I just quoted is further evidence against your claim, as it uses the similar phrase "the Executive and Judicial officers of a State". "The Executive" by itself makes no real sense in this context (however, "the Executive officers of a State" makes perfect sense), but according to the interpretation you offer we must consider it to be a separate thing entirely.
Fourth and finally, the order. In the normal interpretation that everyone else interprets its under, it is "descending", from federal to state. "As a member of Congress [federal], or as an officer of the United States [federal], or as a member of any State legislature [state], or as an executive or judicial officer of any State [state]" has it reasonably start with federal positions, then move onto state positions. Your interpretation instead has it be "as a member of Congress [federal], or as an officer of the United States [federal], or as a member of any State legislature [state], or as an executive [federal or state] or judicial officer of any State [state]." Why start with federal, switch to state, then switch to a combination of federal and state, only to switch back to state again?
So for your idea to work, we have to go with an interpretation that confusingly leaves "executive" by itself with no qualification, doesn't work with the way everything else in the list is listed, goes against grammar rules found elsewhere in the same amendment, and puts the items in a confusing order.