I say 'No': I know people with religious beliefs who I would happily entrust with the welfare of my family and I can think of one atheist that, if we still had a dog, I be worried about asking him to take said dog for a walk.
As a category 'has religious beliefs' is too broad, as is 'has no religious beliefs', to be useful in terms of individuals: and when in comes to public office I'd imagine that factors such as proven competence and relevant experience are more important, and that these aspects are likely to figure more with the electorate and in any appointment process.
In addition, in my experience, those capable of public office or senior public service positions are able to balance their personal views with public responsibilities, and if they don't then there are process where concerns can be raised. I've had experience of one senior NHS manager recusing themselves from discussion and decisions on a matter that involved their personal religious beliefs.
The problem with 'broad brushes' is that everything gets covered in paint irrespective of whether or not painting was required.