I think it clearly does. It applies to any position of public trust in the U.S., be it at the federal, state, or local level. If you look at Article I, Section 6 it says that "No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States which shall have been created, or the Emoluments whereof shall have been increased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office." The second half of that clause has always been interpreted to mean that you cannot hold office in Congress if you are also holding office at any other level. Can't be a state senator and a U.S. senator, or a mayor and a congressman, or a state judge and a city councilman. "Under the United State" in Article VI undoubtedly means the same.
You may be right. But if you’re right, then someone needs to explain how it is that states HAD such tests for office for a long time? Was the principle just not tested in front of SCOTUS? More likely, I think, it was simply not applied to states. Be interesting to have a summary of court thinking on this issue from the 1790s on.