The nation did not disestablish State churches: the States did. For instance Massachusetts disestablishlished its Church in 1835 or so. Ironically, the orthodox Christians did this to keep the Unitarians from taking over. The idea of a state church like that of the Church of England or in the European states, began to be rejected during the First Great Awakening.
The States had many powers that are surprising to modern ears, such as making every determination of citizenship above and beyond the uniform law of naturalization, which was a power expressly enumerated under the Constitution to the Legislature. The Reconstruction era changed all that, with the birth of the Federal leviathan.