It may have been an incorrect interpretation, but it wasn't nonsense. Many people, including at least one who actually helped framed it, believed the no religious test clause implied that the government had general power over religion.
In a February 29, 1788 letter to James Madison, Edmund Randolph, who became the first Attorney General of the United States, wrote,
Does not this exception as to a religious test imply that the Congress, by the general words, has power over religion?
During the Virginia Convention on Ratification of the Federal Constitution, Edmund Randolph, who was a delegate to the general convention that framed the Constitution, made a speech that contained the follow remarks regarding "the [Constitution's] exclusion of the religious test" as "an exception from the
general power of Congress [over religion]"
Freedom of religion is said to be in danger. I will candidly say, I once thought that it was, and felt great repugnance to the Constitution for that reason. I am willing to acknowledge my apprehensions removed; and I will inform you by what process of reasoning I did remove them. The Constitution provides that "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." It has been said that, if the exclusion of the religious test were an exception from the general power of Congress, the power over religion would remain. I inform those who are of this opinion, that no power is given expressly to Congress over religion. The senators and representatives, members of the state legislatures, and executive and judicial officers, are bound, by oath or affirmation, to support this Constitution. This only binds them to support it in the exercise of the powers constitutionally given it. The exclusion of religious tests is an exception from this general provision, with respect to oaths or affirmations. Although officers, &c., are to swear that they will support this Constitution, yet they are not bound to support one mode of worship, or to adhere to one particular sect. It puts all sects on the same footing. A man of abilities and character, of any sect whatever, may be admitted to any office or public trust under the United States. I am a friend to a variety of sects, because they keep one another in order. How many different sects are we composed of throughout the United States! How many different sects will be in Congress! We cannot enumerate the sects that may be in Congress! And there are now so many in the United States, that they will prevent the establishment of any one sect, in prejudice to the rest, and will forever oppose all attempts to infringe religious liberty. If such an attempt be made, will not the alarm be sounded throughout America? If Congress should be as wicked as we are foretold they will be, they would not run the risk of exciting the resentment of all, or most, of the religious sects in America.
-- The Debates in the Several State Conventions on the Adoption of the Federal Constitution ;Elliot's Debates, Volume 3; Tuesday, June 10, 1788.