I think you are right.The problem happens when religion and politics mix. It only produces a toxic, freedom-reducing, poison that kills democracy.
But it's complicated. Joe Biden, a practicing Catholic, is known to have said that his politics are based on his values. And many of his values are based on his Christian faith. The social welfare state in North America and Europe goes all the way back to the Christian humanism and Christian socialism of the 19th century.
I am certain Martin Luther King's advocacy for civil rights laws was rooted in his Christianity. Many of his famous speeches are basically sermons.
So I have been thinking that separation of politics and religion is a more complicated question than a black and white question