Your reading is basically that the first amendment prohibits an otherwise permissible state law because voters support that law based on their belief in Christianity. But it would be fine for voters to support a law based on their belief in say socialism or secular humanism. That’s exactly backward—it singles out religious faith as an improper basis for supporting an otherwise permissible law.

Regardless, that clause is about the federal government not interfering with official state churches (“establishments of religion”) which existed until the 1830s.