Frank, you’re correct here. Politicians are not clergy. They are not supposed to represent their own views but rather the will and view of the people they serve. Thus the term “representative”.
At the same time, if a political figure does claim to be a member of particular religious group then it’d be a good idea to clarify their personal vs their representative role view from time to time to avoid this needless confusion.
Finally, with regard to Trump holding up a Bible in front of a church, I’ve never understood the outrage. What did you want him to hold up? A copy of Sports Illustrated? It’s a symbol of what a Church building stands for, much like a cross. Get over it.