In the 1990s the U.S. Supreme Court decided a handful of religious liberty cases on the basis of the First Amendment’s free speech clause. The most significant of these was Rosenberger v. University of Virginia (1995). In that case, the University of Virginia had denied funding to a religious student publication called Wide Awake. The case began with a focus on the establishment clause, and it might have been based on the free exercise of religion—but it ended up being about free speech.
One’s right to the free exercise of religion does not translateinto the right to government office.