You could probably make it work, a chaplain as a character, although I have my doubts that you could wrap a whole show around one.
I don't think anyone is suggesting have the show be based upon a single character and their career field, just that they would be part of the mix. An aspect of Starfleet/Federation life that hasn't been explored in depth.
The Federation has (hopefully) more than a single mono-culture, religion is one way of peering into a culture and society. The crew of a starship can do more over the course of an episode than "talk shop."
The character of the Chaplain can be used to explore the societies of the various Federation members.