I've read a good chunk of this thread, but not all so if this point has been made already please forgive me.
Many people in America (and I would say maybe world wide) consider the President of the USA to be the most powerful person in the world. I think more people are willing to trust someone who answers to a higher power as something that might keep him in check, whereas someone with no such beliefs would essentially be above the law with no one to answer for. I would surmise that this goes doubly recently where it's been the other branches of government are largely ineffectual.
Not that this in any way makes logical sense, but people like the idea of everyone having to answer to someone.
I'm pretty sure the president answers to the citizens, or at least that's what someone once said....Kennedy maybe? I can't remember. Even if I just imagined it doesn't it sound better that the president answer to the people rather than an imaginary "higher power"?