Also Peter Thiel.
I'm not American so correct me if I'm wrong. But wasn't the first amendment about disallowing the government from establishing or favoring a specific religious establishment. Meaning it would be illegal to say make a law that favors or teaches baptism in school at the expense of another Christian flavor. What I understand of the history behind freedom of religion in the US was that people were sick of the UK mandating the very specific theology of the Anglican church and instead wanting a system where people were allowed to practice any flavor of Christianity that they wanted. From this it would be no problem being a religious person in office but it would not be okay for that person to give benefits or favor to their specific church while in office.
Not making policy based on religious belief would be impossible however. One's religion would impact one's entire morality and value system. So one could say we should enact policy X because the bible says so. But one couldn't say we should enact policy X because the southern baptist Convention just made a ruling that policy X should be enacted and now we have to make a law that matches that ruling.