I am not trying to have the last word.
The "but" is an instrument of language that seperates one set of conditions from another set of conditions in the discussion of a topic.
The sentence with the "but" is meant to sya, essentially: "Yes, satisfy the above you can be president, BUT if these other conditions exist, you cannot be president."
"...reasserting your rock-solid belief that he is ineligible."
Rock-solid? You bet. Doesn't mean someone or some entity won't try to "nuance" it to death.
OK, I think I'm following you now. I agree with your take on the operative effect of the word "but" on the Vice-Presidential eligibility clause. That is not, however, where I see the ambiguity. The ambiguity lies in what exactly the other conditions are that would prevent eligibility to hold the office of the President.