I had not come across this passage (by John Tyler, tenth president of the U.S.) before today:
“The United States have adventured upon a great and noble experiment, which is believed to have been hazarded in the absence of all previous precedent — that of total separation of Church and State. No religious establishment by law exists among us. The conscience is left free from all restraint and each is permitted to worship his Maker after his own judgment. The offices of the Government are open alike to all. No tithes are levied to support an established Hierarchy, nor is the fallible judgment of man set up as the sure and infallible creed of faith. The Mohammedan, if he will to come among us would have the privilege guaranteed to him by the constitution to worship according to the Koran; and the East Indian might erect a shrine to Brahma, if it so pleased him. Such is the spirit of toleration inculcated by our political Institutions.”
So, it's not only Thomas Jefferson who understood the Constitution to be establishing a "separation of Church and State."
Of course, I don't see how the First Amendment and Article VI ("no religious test shall ever be required as a qualification to any office or public trust in the United States.”) can reasonably be interpreted any other way.