Fine, but let's look at all of that section of Article VI:

The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.

Bound by oath to support the Constitution, but no religious test required to assume that duty. Ok cool. Your duty as a government official is to the Constitution, not to your religion, and your religion is immaterial -- and this is key -- provided you uphold the Constitution.

Now let's look at the "establishments clause" of the First Amendment:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof

If you will not uphold the First Amendment's establishments clause, if you are scheming to establish a Christian theocracy, you are not supporting the Constitution. If you are not supporting the Constitution, Article VI in fact disqualifies you from public office.

For that matter, you don't even need the First Amendment to make the point. A theocracy would immediately run afoul of Article VI itself, as it would require officials to adhere to the state's church doctrine.